Languages such that DFA requires $\Omega(c^k)$ states but NFA needs only $O(k)$ states?
The language $L_k$ that you list doesn't work.

There is a straightforward NFA for the language $L_k$ you list, with $k+1$ states.  It guesses where the suffix starts, then checks that the suffix is equal to $s_k$.

In particular, the NFA has states $0,1,\dots,k$, with transitions $0 \to 0$ on each symbol, transitions $i-1 \to i$ on the symbol that matches the $i$th character of $s_k$, and $k$ being an accepting state.

It follows, using the subset construction, that there exists a DFA with $2^{k+1}$ states.

Consequently, that language does not meet your requirements, if $c&gt;2$.