I have proven Hilbert's Theorem 90 for finite extensions, that is for a finite Galois extension of fields $L/K$ with Galois group $G$, $H^{1}(G,L^{\times})=1$.
I'm unsure as to how to proceed to the infinite case. I was hoping that I might be able to use the fact that Galois groups of infinite extensions are profinite groups in some nice way.
Let $L/K$ be an infinite (algebraic) Galois extension. You are completely right about the Galois group $\mathrm{Gal}(L/K)$ being a profinite group. More precisely, $\mathrm{Gal}(L/K)$ is the inverse limit of the groups $\mathrm{Gal}(M/K)$ where $M$ runs over the finite Galois extensions $M/K$, with transition morphisms being the restrictions $\mathrm{Gal}(M/K) \to \mathrm{Gal}(M'/K)$ whenever $M' \subset M$ (see Proposition 2.3.1 in Sharifi's notes).
By proposition 2.2.16 in Sharifi's notes, the first continuous cohomology group of $L^{\times}$ is $$H^1(\mathrm{Gal}(L/K), L^{\times}) \cong \varinjlim\limits_{\substack{M/K \\ \text{finite Galois}}} H^1(\mathrm{Gal}(M/K), M^{\times})$$ where the direct limit is taken with respect to inflation maps. You already know that $H^1(\mathrm{Gal}(M/K), M^{\times})$ vanishes, whenever $M/K$ is finite and Galois. Therefore the direct limit above is trivial, showing that $H^1(\mathrm{Gal}(L/K), L^{\times}) = \{0\}$ as wanted.