I like this question!

Restricted to the finite index subgroup $N$, the representation $\pi$ splits into a direct sum of irreducible representations.

I could not see an easy proof of this, but the proof goes along the following lines.

Suppose $I$ is a totally ordered indexing set and for each $i\in I$, $W_i$ is an $N$ invariant non-zero (closed) subspace of $H$ such that $i<j$ implies $W_i\supset W_j$. Then I claim that the intersection $\cap _{i\in I}W_i$ is a closed non-zero $N$ invariant subspace. Let $P_i$ be the projection map from $H$ onto $W_i$. This map is $N$-equivariant. Consider the finite sum $p_i= \sum _{g\in G/N} gP_ig^{-1}$. Being a $G$ invariant self adjoint bounded operator, by Schur's lemma, $p_i$ is a scalar $c_iI$ for some $c_i\geq 0$. The scalar $c_i\geq 1$: if $v\in W_i$ has norm one, then $c_i=(p_iv,v)\geq (P_iv,v)=1$.

On the other hand, the $W_i$ form a decreasing family. Suppose $w\in H$. The net of numbers $(P_iw,w)$ decreases to $(Pw,w)$ where $P$ is the projection to the intersection $\cap W_i$. (This last statement needs a proof, which is tedious but routine. One has to replace the family $(P_iw,w)$ by a countable subfamily whose lim inf is the lower limit of the family, and then argue that the lower limit is $(Pw,w)$).

From the last two paragraphs, for any $w\in H$ of norm one, we have $c_i=(c_iw,w)=\sum _{g\in G/N}(gP_ig^{-1}w,w)$ decreases to $\sum _{g\in G/N}(gPg^{-1}w,w)$. Since $c_i\geq 1$, it follows that the projection $P$ is non-zero. Hence the intersection $\cap _{i\in I}W_i$ is non-zero.

By the Hausdorff maximality principle, there exists a minimal $N$ invariant non-zero closed subspace $W$ in $H$. This must necessarily be irreducible. The finite sum $\sum _{g\in G/N}gW$ being non-zero and $G$ invariant, is dense in $H$. Each $gW$ is also $N$-irreducible since $N$ is normal. It follows that $H$ is a direct sum of possibly a smaller collection of $gW$.