(This is my first post here, so please excuse me if I'm not following proper etiquette.)
First, I noted that both $K$ and $S$ are subgroups, thus their intersection is a subgroup. We can write that $|G| = p^km$ and $|S| = p^k$ where $p$ is a prime and $m$ has no factors of $p$. By Lagrange's theorem, since the intersection is a subgroup of S, then the cardinality of the intersection must be of the form $p^l$ where $l \leq k$. We can write $K$ as $p^rn$ where $r \leq k$ and $n$ has no factors of $p$. If I can show that $l=r$, then it seems that would be enough to show that the intersection is a sylow p-subgroup of K, but I can't seem to get anywhere else with anything.
I recall a theorem that states that since K is normal, then the intersection of K and S is normal in S but I don't know if that helps me much. There is the corollary of Sylow's 3rd theorem in my book that states that "A Sylow p-subgroup of a finite group G is a normal subgroup of G if and only if it is the only Sylow p-subgroup of G" which seems like I could use that somehow, but I'm not too sure.