  The electronic structure of LiNiO$_2$, a promising Li-ion battery cathode
material, has remained a challenge to understand due to its highly covalent yet
correlated nature. Here we elucidate the electronic structure in LiNiO$_2$ and
the related compound NaNiO$_2$ using x-ray absorption spectra (XAS) and quantum
many-body calculations. Notably, we use inverse partial fluorescence yield to
correctly measure the Ni $L$-edge XAS, which is inaccurate using conventional
methods. We show that the XAS are indicative of a strong Jahn-Teller effect in
NaNiO$_2$ and a bond disproportionated state in LiNiO$_2$, supporting a theory
of a high-entropy, glassy disproportionated state that stabilizes charging
cycles in LiNiO$_2$.
