Prove that the centralizer $C_K(H)$ has finite index in $K$

Let $H,K$ be subgroups of a group $G$. If $[H,K]$ is finite and $H$ is finitely generated, then the centralizer $C_K(H)$ has finite index in $K$. More precisely, if $[H,K]$ has order $n$ and $H$ is $m$-generated, then $|K : C_K(H)| \leq n^m$.

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Hint 1: If $H = \langle x_1, \ldots, x_m \rangle$, then $C_K(H) = \cap_{i=1}^m C_K(x_i)$.
Hint 2: $[K : C_K(x)]$ is the number of conjugates $k^{-1}xk$, where $k \in K$.
Hint 3: Notice that $k^{-1}xk = x[x,k]$.