The two density-functional methods of calculating excitation energies proposed in the preceding paper, combined with the recently formulated quasi-local-density approximation for the equiensemble exchange-correlation energy functional [W. Kohn, Phys. Rev. A 34, 737 (1986)], are applied to the He atom. Although the splittings between nearly degenerate levels with different angular momenta are badly overestimated, in both approaches the averages over angular momentum and spin of the experimental excitation energies measured from the ionization threshold are reproduced within a few percent. The computed self-consistent ensemble-averaged densities and the Kohn-Sham potentials associated with them are discussed.