Writing a program that solves the halting problem is trivially(?) impossible.

However, writing a program that solves the halting problem for some input programs and input inputs, but will sometimes answer "I don't know", is trivially possible.  (For instance, the most trivial solution answers "I don't know" to everything.)

What is the upper bound on how many programs+inputs a program of this sort can give non-"I don't know" answers to?  Specifically, for all Turing machines of length n, consider the function T(n), which is the maximum number of Turing machines that a pseudo-Halting-detector could give non-"I don't know" answers for.  (Essentially, that machines "accuracy").  Is T computable, and can you write an algorithm that computes it?