My question is, do you also know that K(E)? K(K(E))?

I have a sensory/gut experience of being a thinking being, or, as you put it, E.

Based on that experience, I develop the abstract belief that I exist, i.e., K(E).

By induction, if K(E) is reliable, then so is K(K(K(K(K(K(K(E)))))))). In other words, there is no particular reason to doubt that my self-reflective abstract propositional knowledge is correct, short of doubting the original proposition.

So I like the distinction between E and K(E), but I'm not sure what insights further recursion is supposed to provide.