Which functionals should I include for a simple hydrogen atom system?

I am writing my own (orbital-free, for now) 3D density functional theory code from scratch. It currently works correctly for the simple harmonic oscillator potential $$V_\text{ext}(\mathbf r) = \frac12\omega^2 |\mathbf r|^2$$, yielding a ground state energy of $$\frac32\omega$$ (atomic units). I am trying to extend this to find the GS energy of a hydrogen atom.

The SHM code uses the von-Weizsäcker KE functional and the external potential term $$\int d\mathbf r\ n(\mathbf r)V_\text{ext}(\mathbf r)$$. However substituting $$V_\text{ext}$$ for $$-1/r$$ (the potential generated by a proton at the origin) yields an energy of around -12 eV, which is too high. Moreover, unlike for the SHM, the obtained GS energy seems to vary depending on the grid size and spacing, suggesting that I am not including the correct functionals in the energy calculation. Further evidence for this is the energy of the explicit 1s orbital density also being slightly too high (~11 eV).

So my question is: am I using the wrong kinetic energy functional? Should I be adding other terms to the energy expression? What is the setup to correctly obtain the correct GS for the H atom?