Ok, I was checking if there was an obvious flaw that you might have missed. Now I'm trying to actually understand your proof, and I'm getting stuck on the first step, where you invoke the Diagonal Lemma. Now a typical statement of the Diagonal Lemma is:

Diagonal Lemma: If T is a theory in which diag is representable, then for any formula B(x) with exactly one free variable x there is a formula G such that G <=> B(g(G)) is a theorem in T. (g(G) is the Gödel number of G. This was copied from page 2 of http://www.cs.cornell.edu/courses/cs4860/2009sp/lec-23.pdf)

If you look at page 3 of that PDF, it gives an explicit formula for G, which happens to contain an existential quantifier. Now my question is, how do I translate that G into a code string (the one named "box" in your proof)? Also, is this supposed to be obvious, or are you assuming some background knowledge beyond the standard expositions of the Diagonal Lemma and Löb's Theorem?