Proving there exists an infinite number of Turing machines is a decidable problem I have a Turing machine M and I have to answer the question are there infinitely many Turing machines that recognize L(M)? I also need to describe a Turing machine that solves this problem. My idea is to say that we can simply increase the number of states by adding a new unreachable state. Each time I add a new state, it's still considered a new TM and because they're unreachable, I can say that they still accept the same language. 

The only problem I'm having is describing a TM that can solve the problem. I was thinking that if I create a new TM (let's call it N) that represents M, I could say that N just needs to modify the transition function to create an unreachable state and once it does that it can add an infinite number of unreachable states which means there exists an infinite number of TM's. My problem is that I'm not sure if this is all I have to do to prove this and if it is then is my TM N enough to solve the problem or does it need to do more than just modify the transition function? N has to be decidable so it can't just keep adding states or else it will go on forever.
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 $L(M)$.