We present a combined observational and theoretical analysis to investigate the nature of plasma turbulence at kinetic scales in the Earth's magnetosheath. In the first decade of the kinetic range, just below the ion gyroscale, the turbulence was found to be similar to that in the upstream solar wind: predominantly anisotropic, low-frequency and kinetic Alfvén in nature. A key difference, however, is that the magnetosheath ions are typically much hotter than the electrons, ${T}_{{\rm{i}}}\gg {T}_{{\rm{e}}}$, which, together with ${\beta }_{{\rm{i}}}\sim 1$, leads to a change in behavior in the second decade, close to electron scales. The turbulence here is characterized by an increased magnetic compressibility, following a mode we term the inertial kinetic Alfvén wave, and a steeper spectrum of magnetic fluctuations, consistent with the prediction ${E}_{B}({k}_{\perp })\propto {k}_{\perp }^{-11/3}$ that we obtain from a set of nonlinear equations. This regime of plasma turbulence may also be relevant for other astrophysical environments with ${T}_{{\rm{i}}}\gg {T}_{{\rm{e}}}$, such as the solar corona, hot accretion flows, and regions downstream of collisionless shocks.