I can see that the question wants us to go through the sort of procedure you describe. I consider the method analogous to finding the critical mass for a neutron star. One calculates the total kinetic energy of a fermi gas, and adds a potential term, the sum of which is then minimised.
My problem is that whilst the kinetic term is a function of electron density, the potential term is not. So if 'Determine the electron density that minimizes the total energy (electrostatic plus
kinetic) per ion.' I will find that the minimum is at zero density.