Q. What solid evidence is there for the divine inspiration of the New Testament other than alleged statements by Peter and Paul? Who gave them the authority to make such a declaration of divinity?
The divine inspiration of the New Testament is something that Christians accept by faith. If we believe in Jesus, we believe in the writings that testify about Jesus. That does not require the same kind of evidence that something would that we wanted to accept by reason.
Nevertheless, something we accept by faith should be reasonable. Faith and reason are two complementary ways of knowing the truth. While each understands a different aspect of the truth, their findings should be compatible.
And we can observe that the New Testament provides a reasonable account of how the trajectory of God’s redemptive work traced in the Old Testament reaches an intrinsically appropriate culmination in the life, teachings, sacrificial death, and resurrection of Jesus and in the way of life of the community of his followers. In other words, the New Testament reasonably is what we would expect it to be if it were divinely inspired. That does not convince us that it is, it reassures us that it is, once we already accept that by faith.
We would also expect the New Testament writers to be aware of this inspiration, and they are, as you note. And if the New Testament is indeed inspired, then that is what gives the writers the authority to declare that it is. Admittedly that is circular. But as one of my theology professors once said, “The only way to do theology is in a circle. The issue is how you get onto the circle.”
For many people, the problem in getting onto the circle is recognizing that faith is indeed as legitimate a way of knowing as reason. Along those lines, I find the following quotation from Blaise Pascal helpful. He was one of the most brilliant mathematicians who ever lived, so he certainly knew how to think reasonably and logically. But his genius also gave him the insight to realize that, as he put it, “The ultimate task of reason is to recognize that there is an infinite number of things that surpass it.” Specifically, there are divine realities that surpass human reason but that we can nevertheless access through the faculty of faith.
Now the capacity for faith is also the capacity for doubt. Anything that must be known by faith can also be doubted. But that is not a bad thing. By working through our doubts, we strengthen our faith. There is a difference between doubt and skepticism. Skeptics begin with the stance that they are not inclined to believe. People who doubt want to believe.
Your question is certainly a legitimate one. I hope you will continue to pursue it, and I hope you will do so as a doubter, not as a skeptic.
Good Question 
Excellent explanation