I found a natural proof.
Consider the multiplicative group $L^*$ as a locally compact group (see Lemma 3 in my other answer for continuity of inversion) and let it act on the space of norms on $L$ (considered as a $K$ vector space) by $\|\cdot\|\mapsto |N(x)|^{1/n}\|x\cdot\|$ for $x\in L^*$. I claim that there are $K^*$-fixed norms on $L$. Indeed, for every given norm on $L$, the corresponding operator norm on $L$, considered as an algebra of operators for the regular action, is $K^*$-fixed. The compact group $L^*/K^*$ (homeomorphic to $\mathbb{P}^{n-1}(L)$) acts on the space of $K^*$-fixed norms. By averaging with respect to the Haar measure we find a fixed point. Normalized it to have $\|1\|=1$, this fixed point must be $|N(\cdot)|^{1/n}$. It follows that the latter is a norm.