Proving there exists an infinite number of Turing machines is a decidable problem
Given $L(M)$, there are countably infinite number of Turing machines that accept this language. You already describe in your post how to generate these machines. Thus the problem 


  are there infinitely many Turing machines that recognize L(M)


is decidable. The TM machine that decides it simply outputs $1$ (ACCEPTS) on any input $\langle M \rangle$. 

Note that your TM $N$ does not need to construct all machines, it just needs to always response "YES" since for any $M$ indeed there exists infinite number of TMs accepting $L(M)$.