We present polarization dependent x-ray-absorption spectra at nickel L edges of well-characterized La₂NiO_4+δ single crystals. In the stoichiometric compound the splitting between the x²−y² and the 3z²−r² orbitals is 0.7 eV, according to a fit of the 2p⁵3d⁹ multiplet to the spectra. This value is in agreement with an assignment of dd excitations of the optical spectrum. The Ni L edges of the doped compound are consistent with the isotropic prepeak observed at the oxygen 1s edge. Theory does not predict holes on the apex oxygens, but we argue that doping causes a polaronic deformation which reduces the tetragonal distortion of the NiO₆ octahedra, and delocalizes the hole over all six ligands.