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@Discipulus_Didicit
Were those human writters ever been wrong about anything?
Yes. Recall Paul's error regarding his view of Jesus prior to his conversion. But the authority of the Bible is not based on whether the human authors are capable of error. The authority of the Bible is based on whether or not the words are true. It is the written message recorded in the Bible that is inerrant, not the human authors.
So we know that the books were inspired by God because the human writers wrote a passage saying that it was inspired.
In a sense, yes. However, that would be an incomplete statement. It is not only because the human writers wrote a passage saying it was inspired. Voddie Baucham gave an extremely concise statement that would give a more complete picture of how we know the Bible is the Word of God:
"I choose to believe the Bible because it is a reliable collection of historical documents written down by eyewitnesses during the lifetime of other eyewitnesses. They reported supernatural events that took place in fulfillment of specific prophecies and claimed that their writing are divine rather than human in origin."
I italicized that last bit to show there is far more than a single human claim in determining how we know the Bible is inspired. But I suppose much of the disagreement would probably come down to the question of how we can know anything at all.