Returning to the proof of the main theorem: uniqueness of $L$ follows because an isomorphism $\ell \cong k_{L'}$ over $k_K$ between unramified extensions lifts to a $K$-isomorphism $L \cong L'$, and $[L:K] = [L':K]$ forces this to be an isomorphism of the full extension. For the Galois property: \begin{align*} |\operatorname{Aut}_K(L)| = |\operatorname{Aut}_{k_K}(k_L)| = [k_L:k_K] = [L:K], \end{align*} so $L/K$ is Galois. The natural map $\operatorname{Gal}(L/K) \to \operatorname{Gal}(k_L/k_K)$ given by reducing automorphisms modulo $\mathfrak{m}_L$ is the bijection from the lemma, hence an isomorphism of groups.