Turing machine enumerator
Indeed, there is such a Turing Machine. First, realize that it's computable to convert any integer $k$ into the $k$th Turing Machine description, just by enumerating them. Then, once we have the Turing Machine description, we can use a Universal Turing Machine to compute the function described.

The key is that, while there are less states in the $n$-state machine than in one with $n^2$, any number of additional states can be encoded on the tape of the Turing Machine. So there's no limit to the amount of information we have when computing the $k$th function, since an arbitrary amount of information can be encoded on the tape.