If $K$ is 'everything 1 mod N' for some N, then the canonical model of $\mathbf{Q}^\times_+ \backslash \mathbf{A}^\times / K$ is exactly $\mu_N / \mathbf{Q}$, the scheme of $N$-th roots of unity. Any open compact $K$ will contain one of these, so $GL_1 / \mathbf{Q}$ Shimura varieties all look like quotients of $\mu_N$ for some $N$. 

Hence the answer to "How do we finish the proof from there on to get Artin's reciprocity?" is "we prove the Kronecker--Weber theorem" [i.e. every abelian extension of $\mathbf{Q}$ is contained in a cyclotomic field].

Perhaps it's disappointing that Shimura varieties don't tell you how to prove Kronecker--Weber. But they do something much more important: they tell you how to *generalize* Kronecker--Weber, pointing you towards a much more general (mostly conjectural) picture of which Kronecker--Weber is just one small part.