Steve Ray has a list of 35 loaded Questions for "Bible Christians" (quotation marks his)(link to the whole list). This is number 11/35. I'm trying to provide the answers in a common format, for easy reference.
11) How do we know, from the Bible alone, that the individual books of the New Testament are inspired, even when they make no claim to be inspired?
We accept them on faith.
1) We know that the whole Old Testament was inspired, even though many of those individual books don't make any such claim.
2) And, of course, it is sufficient for us that 2 Timothy tells us that all Scripture is inspired (and note that 2 Timothy doesn't seem to think that it is necessary to attach an infallible and inspired table of contents of Scripture).
2 Timothy 3:16-17
All scripture is given by inspiration of God, and is profitable for doctrine, for reproof, for correction, for instruction in righteousness: that the man of God may be perfect, throughly furnished unto all good works.
3) The fact that we accept them on faith doesn't mean that our faith is irrational. Thus, for example, the following are also the case.
a) The Holy Spirit testifies to the inspiration His word.
b) We have historical evidence that also supports our conclusion that a particular book or passage is inspired Scripture.
c) But ultimately, "my sheep hear my voice" is the governing principle.