  We present a kinetic-energy density-functional theory and the corresponding
kinetic-energy Kohn-Sham (keKS) scheme on a lattice and show that by including
more observables explicitly in a density-functional approach already simple
approximation strategies lead to very accurate results. Here we promote the
kinetic-energy density to a fundamental variable along side the density and
show for specific cases (analytically and numerically) that there is a
one-to-one correspondence between the external pair of on-site potential and
site-dependent hopping and the internal pair of density and kinetic-energy
density. Based on this mapping we establish two unknown effective fields, the
mean-field exchange-correlation potential and the mean-field
exchange-correlation hopping, that force the keKS system to generate the same
kinetic-energy density and density as the fully interacting one. We show, by a
decomposition based on the equations of motions for the density and the
kinetic-energy density, that we can construct simple orbital-dependent
functionals that outperform the corresponding exact-exchange Kohn-Sham (KS)
approximation of standard density-functional theory. We do so by considering
the exact KS and keKS systems and compare the unknown correlation contributions
as well as by comparing self-consistent calculations based on the mean-field
exchange for the keKS and the exact-exchange for the KS system, respectively.
