Reference: the number of smooth structures on a topological $n$-manifold, $n\geq 5$ is finite

The statement you refer to is a theorem of Kirby and Siebenmann; see this answer for more information.

As for your last statement, the number $k$ is not always equal to the number of smooth structures on $S^n$. For $n \geq 5$, the number of isomorphism classes of smooth structures on $T^n$ is in one-to-one correspondence with $H^3(T^n; \mathbb{Z}_2)$; see this MathOverflow answer. Therefore $T^n$ has $2^{\binom{n}{3}}$ smooth structures, while the number of smooth structures on $S^n$ is much more complicated, as discovered by Kervaire and Milnor; see here and here.

For example, $T^5$ has $2^{10} = 1024$ different smooth structures, while $S^5$ has only one.