Uniqueness of the direct sum complement of a vector subspace

The error lies in assuming that the only possibility for the supplementary subspace is $\langle v_{m+1},\ldots,v_n\rangle$. It isn't. If $V=\mathbb{R}^2$, $U=\langle(1,0)\rangle$ and if you complete the basis, which vector will you take? You will choose $(0,1)$ probably. And indeed $\langle(0,1)\rangle$ is a supplementary subspace. However, $\langle(1,1)\rangle(=\langle(1,0)+(0,1)\rangle)$ will also work, for instance.