The question is "what is knowledge?", not "do we know that we know p?". And I see no issue with the definition of knowledge as justified, true belief. Now, if I believe p, and you ask me whether I know p, I may say yes. But whether I actually know p will depend on whether my justification is valid (that it really is a justification and a sufficient one) and whether it is true, which has nothing to do with whether anyone knows whether the justification is valid and the belief is true. It's a separate question, and conflating the two questions leads to an infinite regress of skepticism. So the definition of knowledge qua knowledge still stands.
I would also suggest you try to apply your general approach to the very theory you are proposing. I see an opportunity for retorsion arguments.
