Let $\bar{E}$ be the norm completion of $E$, which is a Banach space. Then we can consider $E$ as a dense linear subspace of $\bar{E}$, where the subspace topology and the norm topology on $E$ coincide. In particular, since this topology is homeomorphic to $F$, it is completely metrizable, so $E$ is a $G_\delta$ in $\bar{E}$ (Kechris, *Classical Descriptive Set Theory*, Theorem 3.11). As a dense $G_\delta$, in particular $E$ is comeager in $\bar{E}$. If $x \in \bar{E} \setminus E$, then $E, E+x$ are disjoint comeager subsets of $\bar{E}$, which is absurd by the Baire category theorem. So $E = \bar{E}$ and thus the norm on $E$ is complete.

I think I did not use separability anywhere.