First, I'm assuming we're dealing with finite dimensional spaces. Note that $V\cap W \subseteq V$ so $k \le n=\dim(V)$.
If $V=span\{u_1,\ldots, u_k\}$, then we're done. If not, take $v_1\in V-span\{u_1\,\ldots,u_k\}$ and add it to the list.
If $V=span\{u_1,\ldots, u_k,v_1\}$, we're done. If not, take $v_2\in V-span\{u_1,\ldots,u_k,v_1\}$ and add it to the list.
Continue in this fashion. Now, you should try to answer two questions:
(1.) Why does this process stop?
(2.) Is the resulting set still linearly independent?
But notice that it didn't matter too much that we used $V$ here; we can do the same process for $W$. In fact, the process we used is something we can do in general: linearly independent sets are contained in (and thus can be extended to) a basis.