Within exact electron density-functional theory, we investigate Kohn-Sham (KS) potentials, orbital energies, and noninteracting kinetic energies of the fractional ions of Li, C, and F. We use quantum Monte Carlo densities as input, which are then fitted, interpolated at noninteger electron numbers N, and inverted to produce accurate KS potentials vNs(r). We study the dependence of the KS potential on N, and in particular we numerically reproduce the theoretically predicted spatially constant discontinuity of vNs(r) as N passes through an integer. We further show that, for all the cases considered, the inner orbital energies and the noninteracting kinetic energy are nearly piecewise linear functions of N. This leads us to propose a simple approximation of the KS potential vNs(r) at any fractional electron number N which uses only quantities of the systems with the adjacent integer electron numbers.