Hartree-Fock vs. DFT vs. Hohenberg-Kohn vs. Kohn-Sham
The Kohn-Sham approach is one on top of the original DFT idea proposed by Hohenberg-Kohn.
They propose – to find the energy functional and its equations, which, as you said, are unknown – to start with what is known: the functional if there is no interactions. So that is what they do: approximate the kinetic energy and Slater determinant from the known, non-interacting system and add all the terms that are additionally known (i.e. Coulomb, kinetic energies and the electron/nucleus potential) until you are left with the Kohn-Sham equation, which has &quot;the smallest unknown&quot;, the exchange &amp; correlation potential.
I recommend to read more in a book or on Wikipedia in case you want a more detailed picture on any/all of the methods.