[en] More than 200 low-lying and moderately high-lying doubly excited states (^1.3S^e, ³P^o, ^1.3D^e, ^1.3F^o, ¹G^e) of He-isoelectronic systems (Z = 2-5) have been studied in the nonrelativistic Hohenberg-Kohn-Sham density-functional framework by incorporating the nonvariational work-function exchange potential suggested by Harbola and Sahni and a simple parametrized local Wigner-type correlation functional. In essence, a Kohn-Sham-type equation has been solved numerically. Our results on excited-state energies, excitation energies and differences between excitation energies are in good agreement with available experimental and theoretical results. In 50% of cases, the percentage deviation in the calculated excitation energy is less than unity. The estimates of the relatively small distances (0.02-0.1 au) of the He states below the corresponding N = 2, 3 4 ionization thresholds are in error by 0.5-34.9%, indicating that the Wigner correlation functional needs to be improved for greater accuracy. The excited radial densities display expected structures. Some new states are reported. (author)