Total charge in electron cloud in a hydrogen atom

Essentially, the problem I’m trying to solve is

The
potential at a distance $r$ from the nucleus that is generated by an electron in a
hydrogen atom is given by:
$$V(r) = frac{q}{4pivarepsilon_o}left(frac{e^{-2r/a}-1}{r}+frac{e^{-2r/a}}{a}right) $$
where $a$ is a constant (which I think is the radius of the atom). Use Gauss’ law to calculate the total charge in the electron cloud.

To use Gauss law for a spherical surface (in the form $Ecdot 4pi r^2 = Q/varepsilon_o$), we need field $E = -frac{dV}{dr}$. If we evaluate the field $E$ at $r=a$, so that we contain the entire cloud, we get that
$$E(r) = frac{q}{4pivarepsilon_oa^2e^2}(e^2-5).$$

And I think that something is incorrect – the electron cloud should have charge $-q$, but there’s no way to get that from the above. What am I missing?

(all similar questions on this site are with charge density $rho$ instead of the potential, so they are not useful).