Understanding definition of NP and co-NP
Nondeterministic machines are only allowed to behave nondeterministically in a limited way: very briefly, "Accept iff some branch has [property]" is permitted but "Accept iff no branch has [property]" is not. Unlike with deterministic machines, there's a fundamental element of asymmetry here. Your $N'$ is not, in fact, a nondeterministic Turing machine. 

Rather, it's a "co-nondeterministic Turing machine" (if that were a term). Unsurprisingly, co-NP is exactly the set of languages accepted by some co-nondeterministic Turing machine, and the fact that NP=co-NP is open means exactly that it is not currently known whether nondeterministic and co-nondeterministic machines can compute the same things in polynomial time.