You wrote: "For he is the Knower, and the Knower can know other things, but cannot make Himself the object of His own knowledge, in the same way the fire can burn other things, but cannot burn itself."

It's a Kantian problem. Although Kant resolved it in a very bad way, stating that the foundation of knowledge is the fact that we think, classical philosophy already resolved this much earlier and better. But leaving that aside and using only the Bible, in this excerpt from 1 Corinthians, there is a tacit affirmation that it is possible to examine ourselves to the point of being able to confess ourselves satisfactorily to God. This would not be possible if we could not become objects of our own knowledge. When we say "I", however, we do not say it completely because we have no foundation in ourselves. We managed to transcend a little. Something like thinking beyond ourselves by placing ourselves above the universe, but that's just an exercise in imagination. God, however, is the ultimate foundation. He is the only one who IS in himself. So, by analogy, we can know ourselves, and he certainly knows himself.