Let $E_c$ be the *complete Erdös space* (Erdös, Annals of Math vol 41 1940), defined as the subspace of $\ell^2(\mathbb{N})$ where all coordinates are irrationals. It is polish (separable and completely metrizable) and totally disconnected, but admits a "connectification" namely a (still polish) topology on $E_c\cup\{p\}$ that makes it connected (and of course induces the one on $E_c$). The crucial point is the fact that any nonempty closed and open subset of $E_c$ is unbounded. Then, as in Bill Johnson's answer $E_c$ is the union of the closed and totally disconnected subspaces $\overline{B}(0,n)\cup\{p\}$, $n\geq 1$. It remains to remark that, like any polish space, $E_c\cup\{p\}$ embeds as a closed subset of $H$ (as remarked in Gerald Edgar's answer).