Think of it this way : by the local Kronecker-Weber, $K$ is the maximal abelian extension of $\mathbf{Q}_p$. Now, the the extension $\mathbf{Q}_p(\root{p^m}\of u)$ need not even be galoisian over $\mathbf{Q}_p$ for some appropriate $u\in\mathbf{Z}_p^\times$, so it cannot be contained in $K$.

Think of it this way : $K$ is an abelian extension of $\mathbf{Q}_p$. Now, the the extension $\mathbf{Q}_p(\root{p^m}\of u)$ need not even be galoisian over $\mathbf{Q}_p$ for some appropriate $u\in\mathbf{Z}_p^\times$, so it cannot be contained in $K$. (As it happens, $K$ is the maximal abelian extension of $\mathbf{Q}_p$ by the local Kronecker-Weber theorem, but this fact is not used in the above argument.)