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By definition, any homomorphism from $G_K$ to an abelian group factorizes through $Gal(K^{ab}/K)$; by the local class field theory the latter group is the completion of $K^\times$. Hence it suffices to calculate $Hom (\overline{K^\times},L^\times)$. Lastly, one can note that up to torsion $K^\times\cong \mathbb{Z}_p^{[K:\mathbb{Q}_p]+1}$ and $L^\times\cong \mathbb{Z}_p^{[L:\mathbb{Q}_p]}\bigoplus \mathbb{Z}$ (via the corresponding logarithms).