The standard response to this is that you can’t really use a conclusion as proof of itself. I agree, yet it happens. Rather than address the logical fallacy, I’d like to look at whether the Bible actually makes the claim of infallibly or inerrenacy at all.
Firstly, the obvious. The Bible is not a book so much as a collection of books. As such, it is insufficient for a single book to claim divine status of itself to make the claim that the whole collection makes such a claim. There would need to be a verse that makes the claim of the entire collection of works within the collection of works in order to claim that the Bible claims divine status of the whole Bible.
Let’s look at some verses which are frequently cited in this discussion:
“the words of the LORD are flawless” (Psalm 12:6)
This verse is part of a song about God making a promise to save some people because they were groaning with sufficient volume. There is nothing in this verse that even remotely suggests that it is talking about a written text.
“Your word, LORD, is eternal; it stands firm in the heavens.” (Psalm 119:89)
Again, this is referring to a direct word from God and mentions nothing about a text. This psalm does at least link the word of God with the Law of Moses. We now have a tenuous link of one song within Psalms supporting elements of Exodus and Deuteronomy… or at least supporting the texts referenced in those books.
“Every word of God is flawless” (Proverbs 30:5)
Agur is addressing Ithiel with words of advice to listen to God’s voice. Given the choice of name (Ithiel meaning: God is with me) it seems that the advice is for those in relationship with God to trust his words when they hear them. It also states not to add to them. Again, this is talking about direct communication from God and makes no mention of existing texts.
“All Scripture is God-breathed and is useful for teaching, rebuking, correcting and training in righteousness” (2 Timothy 3:16)
This is probably the most commonly cited verse on the topic of Biblical infallibly/inerrancy Translators sometimes use the term “inspired” instead of “God-breathed”. It really depends on whether they want to encourage a figurative or literal interpretation in their English readers. This is probably the strongest verse to state the case but it is important to remember that the reference to “scripture” is not talking about the collection of books we now call “The Bible” but most likely what we now refer to as the Old Testament. Possibly just the first five books.
So we do have a book in the Bible which claims divine inspiration (not inerrancy) for a part of the Bible which does not include itself. This is important to identify we don’t yet have any internal reason to grant the label of divinely inspired to the speaker, so his testimony of divine inspiration for the older books is, at this stage, merely a human opinion.
“If anyone thinks he is a prophet or spiritual, let him recognize that the things which I write to you are the Lord’s commandment.” (1 Corinthians 14:37)
This seems to be the strongest claim in any of Paul’s writings that could be interpreted as him believing his own words to be the words of God. This demonstrates Paul’s gift of manipulative writing. He is essentially making it a test of a true prophet that the prophet endorses Paul’s teachings as divine commands.
“For no prophecy was ever produced by the will of man, but men spoke from God as they were carried along by the Holy Spirit.” (2 Peter 21)
This is speaking specifically of The Prophets, a series of writings in what we currently refer to as the Old Testament, though the statement itself could be applied to prophets in general. This verse is explaining that the prophets themselves did not know that the prophecies they were making were actually about Jesus and that nobody would be able to make the connection until after the events had occurred. I’ve spoken about the problems with that kind of prophecy in other posts. For now, let’s simply observe that this text is referring to a specific group of texts of which it is not itself included as even in the most liberal interpretation of the text, it does not claim itself to be prophetic.
“For I testify unto every man that heareth the words of the prophecy of this book, If any man shall add unto these things, God shall add unto him the plagues that are written in this book: And if any man shall take away from the words of the book of this prophecy, God shall take away his part out of the book of life, and out of the holy city, and from the things which are written in this book.” (Revelation 22:18-19)
A great threat to end your compilation with. This verse is frequently taken to refer to the entire Bible. Keep in mind again that the Bible was not even an idea at the time that this prophecy was written. It is possible to interpret the 2 Peter verse as supporting this book because it is “prophecy”, but only in the sense that the writer of Revelations doesn’t really know what he is talking about.
To summarise:
Old Testament verses refer to the direct spoken word of God and possibly to the Law of Moses as perfect, infallible and without error.
New Testament verses refer to The Torah and The Prophets (but not psalms or proverbs) as divinely inspired. Paul invites prophets to endorse his writings as commands from God as a test of their powers, and Revelations endorses itself.
Conclusion:
There are numerous books within both testaments which have no text claiming divine authorship or inspiration for that text, including all four Gospels. As such, any claim that the entire Bible is the Word of God, infallible, without error, or any other divine features must be supported by evidence external to the Bible. That external source would have to be infallible and without error to do so.
TL;DR:
While individual authors make several claims about their own and others’ writings, The Bible does not say that the Bible as a whole is infallible and without error.
Church of Jaminology 
Actually, you could argue that this concept has implicit grounding in scripture even if it isn’t explicit – cf John 14:17&26. But I guess its the extent to which we define the doctrine – the verses you’ve cited, particularly 2 Tim 3:16, could be regarded as expressing the doctrine to some extent. I think the way some christians practically use it and and express it in a wider sense may be somewhat based in scripture, but has gone beyond it in certain aspects – for instance the concept of the canon and also saying that the original autographs were truly infallible, but the scriptures we have now may contain some limited copying or translation errors. This is worringly similar to what the pharisees did with the mosaic law – trying to fill what they say as logical gaps in the text with their own reasoning and then elevating that reasoning to dogma. On balance I would tend to agree with you that “the Biblical Infallibility belief is not itself Sola Scripture” – at least how the doctrine is commonly expressed anyway.
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Further to this, developing the doctrine of the divine inspiration of the Bible as we (orthodox christians) have it today is done through the additional step of faith in believing that the Holy Spirit oversaw the preservation of the canonical scriptures (as individual books, or sub collections of books) to the point of guiding the elders of the Church – men who had experienced the new birth and were themselves led by the Holy Spirit – to come to agreement and codify what the church as a whole could recognize as being scripture. In conclusion, I would have to agree with you that it is impossible to logically prove the infallibility of the bible by merely examining the text and making deductions on what it says, but if you “follow the recipe” that it contains, you will come to a point of certitude about the issue (not via rational deduction, but by the witness of the Holy Spirit in the context of a changed life). An analogy is whether a DNA molecule is an ‘infallible’ representation of some other organism – even if you knew everything about genomics that it was possible to know, you would not be able to answer this question properly by merely sequencing the DNA and mapping it as a genome (even if you could somehow assure yourself that each individual gene was valid and consistent with the others according to the best available knowledge) – you would actually need to take the step of placing it within a viable vessel (eg an already living cell) for it to express itself.
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Since the concept that the Holy Spirit oversaw the preservation of the canonical texts (and presumably aided in determining the accepted canon in its various forms throughout history) does not appear in the canonical text itself, would you agree that the Biblical Infallibility belief is not itself Sola Scripture?
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Interesting, but incomplete: you need to consider the implications of 2 Peter 3:16 where Peter characterises the writings of Paul as scripture; and getting more to the heart of the matter are John 5:39 (that the real point of the scriptures is that they testify about Jesus), the gospel message (in a nutshell, that Jesus Christ is the perfect revelation of the invisible Creator God given in atoning sacrifice in order that sinful humanity can be redeemed from bondage to sin through a new birth by the Holy Spirit according to the image of Christ) & 1 Cor 2:14 (which along with John 3:8 teaches the impossibility of coming to a proper understanding of scripture without experiencing the new birth). This then becomes the point you are looking for – a transition beyond mere texts to an external witness than can actually validate the texts: the Living presence of the indwelling Holy Spirit. It is He that leads the reader not merely to a correct understanding of scriptural texts, but an experential awareness that the Bible is in fact “…The Word of God…alive and powerful…expos(ing) our innermost thoughts and desires” (abbr. Hebrews 4:12 NLT). Of course it requires a step of faith to get to that point, but if you pay attention to the text, it should come as no surprise that that would be what’s required to gain any certitude about such an issue.
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Thanks, I’ll look into adding the verses you’ve mentioned as possible internal arguments for biblical infallibility.
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