I am having trouble seeing how class field theory implies the following isomorphism. Let $F_n=\mathbb{Q}(\mu_{p^{n+1}})$, let $L_p/F_n$ be the maximal unramified abelian $p$-extension (i.e. the $p$-Hilbert class field), and let $M_p/F_n$ be the maximal abelian $p$-extension unramified away from $p$. Furthermore, let $L/F_n$ be the Hilbert class field and let $M/F_n$ be the maximal abelian extension unramified away from $p$. Let $\mathfrak{p}$ be the unique prime of $F_n$ dividing $p$ and let $K_n$ be the completion of $F_n$ at $\mathfrak{p}$. Let $U_n$ be the unit group of $K_n$, $U_n^1$ be the principal units of $K_n$ and let $E_n$ be the closure of the global units in $U_n$ and likewise for $E_n^1$. In several sources I've seen the isomorphism \begin{align*} U_n^1/E_n^1 \cong G(M_p/L_p) \end{align*}
and the reference for this was corollary 13.6 in Washington's Introduction to Cyclotomic Fields which states that \begin{align*} U_n/E_n \cong G(M/L) \end{align*} I'm having some trouble seeing how I'd get from the latter to the former. My thought was to tensor both sides with $\mathbb{Z}_p$ since $U_n^1/E_n^1 = (U_n/E_n) \otimes \mathbb{Z}_p$ but I don't see how the analogous statement is true for the Galois groups since the former is a quotient of a subgroup of the latter.
A second related question is the following: If we let $J$ denote the idele group of $F_n$, and $U=\prod_{\mathfrak{q}}U_{\mathfrak{q}}$ then the Artin map gives the isomorphism $J/F_n^{\times}U \cong G(L/F_n)$. Is there an analogous description for $G(L_p/F_n)$? We can obviously write this latter group as a quotient of the former but can we describe the subgroup $H \cong J$ for which $J/H \cong G(L_p/F_n)$ explicitly?
Any help would be very much appreciated.