  The electronic structure of a LaNiO$_3$ bilayer grown along the [111]
direction and confined between insulating layers of LaAlO$_3$ is theoretically
investigated using a combination of first principle calculations and effective
multi-orbital lattice models. The LDA band structure is well reproduced by a
tight-binding model for the Ni-$e_g$ orbitals defined on the buckled honeycomb
lattice. We highlight peculiar properties of this model which include almost
flat bands as well as linear and quadratic band crossing points. The effect of
local correlations is discussed within the LDA$+U$ scheme and within the
Hartree-Fock approximation for interacting multi-orbital lattice models. Over a
wide range of interaction parameters we find that a ferromagnetic phase is
energetically favored. We discuss the possibility of additional orbital order
which could stabilize a spontaneous Chern insulator with chiral edge modes or a
staggered orbital phase with a $\sqrt{3}\times\sqrt{3}$ reconstruction of the
unit cell. By studying an interacting nickel-oxygen lattice model we find that
the stability of these orbitally ordered phases also depends on the value of
the charge-transfer energy. Controlling the charge-transfer energy might
therefore be an important step towards engineering exotic electronic phases in
certain classes of oxide heterostructures.
