The kinetic energy term could easily be calculated by using the same principle as LDA. That is, we could use the result derived from a homogeneous electron gas. This is the approach followed by Thomas Fermi theory. Unfortunately, the accuracy needed to describe the small energy changes that characterise chemical bonding is not sufficient with this approach. We need another way of getting the kinetic energy.
Two clever chaps, Kohn and Sham introduced a set of orbitals from which the electron density can be calculated. These Kohn-Sham orbitals do not, in general, correspond to the actual electron orbitals. Likewise, the Kohn-Sham eigenvalues are not in general the same as the real energy levels. The only connection the Kohn-Sham orbitals neccessarily have to the real electronic wavefunctions is that they both give rise to the same charge density. The Kohn-Sham orbitals are used to calculate the kinetic energy. The property of the orbitals that makes them useful in the derivation is their orthonormality. The tricky problem of a system of interacting electrons has now been mapped onto that of a system of non-interacting electrons moving in an effective potential.