Find the smallest set of strings which "covers" a given set of strings (coverage = containing as substring)
Without loss of generality, we can take $l=k$.  In particular, any solution to $T(k,l)$ can be converted into a solution for $T(k,k)$ by shortening each string that's longer to $k$ to a string of length $k$ (by deleting as many characters from the beginning or end as needed to get down to length $k$, chosen arbitrarily).  So, it will simplify your life to focus on solving $T(k,k)$, i.e., finding a set of strings of length $k$.

Here is one possible approach.  You don't need to exhaustively generate all strings of length $k$.  Instead, you can generate all length-$k$ substrings of words of $S$.  If each word of $S$ has length $\le n$, then there will be $\le (n-k+1) \cdot |S|$ of them.  Then, you could apply some standard set-cover algorithm to that.  So, you don't need to generate exponentially many strings; just polynomially many.

It's possible there may be better algorithms that take into account the combinatorial structure of the strings; I don't know.