The Hohenberg–Kohn theorems[1] guarantee that, for a system of atoms, there exists a functional of the electron density that yields the total energy.
The lack of the true exchange–correlation functional is a well known problem in DFT, and there exists a huge variety of approaches to approximate this crucial component.
One of the first attempts to do this (even before the formulation of the Hohenberg–Kohn theorem) was the Thomas–Fermi model, which wrote the kinetic energy as[2] This expression is based on the homogeneous electron gas and, thus, is not very accurate for most physical systems.
Finding more accurate and transferable kinetic-energy density functionals is the focus of ongoing research.
By formulating Kohn–Sham kinetic energy in terms of electron density, one avoids diagonalizing the Kohn–Sham Hamiltonian for solving for the Kohn–Sham orbitals, therefore saving the computational cost.