Prove that $N \subseteq H$

The elements of $G/N$ are of the form $gN$ for $g\in G$, and the identity element of $G/N$ is $N$.

Easy way to do it: notice that $H=\pi^{-1}(K)$ and that $N=\pi^{-1}(\{N\})$ and that $\{N\}\subset K$.

If you want to do it more by "hands", take an element $n\in N$. To show that $n\in H$, you must show that $\pi(n)\in K$. But $\pi(n)=nN=N$.