Kronecker-Weber à la Wiles

Question

I've been studying the notes of E.Kowalksi on the Course by Tunnell on Wiles's proof of Taniyama-Shimura. As a baby case, they reprove the Kronecker-Weber Theorem. However, I'm confused by the last step, which is on page 19 of https://people.math.ethz.ch/~kowalski/tunnell-2.pdf. At this point, we have already shown that every Galois representation is modular, i.e. there exists a Dirichlet character $\chi$ such that $\rho=\rho_\chi$. Then we pick a finite abelian galois extension $K/\mathbb{Q}$ with Galois group $\mathbb{Z}/p^n$ and associate a Galois representation via $$G_{\mathbb{Q}}\rightarrow Gal(K/\mathbb{Q})=\mathbb{Z}/p^n\rightarrow \mathcal{O}_{\mathbb{Q}_p(\mu_{p^n})}^\times$$ by choosing a primitive $p^n$-th root of unity. Now we know that $\rho$ is modular. How can we now conclude that $K\rightarrow \mathbb{Q}(\mu_{p^n})$? I feel that I'm missing something very elementary here, but I just can't see it.