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We study the effects of inelastic dark matter self-interactions on the internal structure of a simulated Milky Way (MW)-size halo. Self-interacting dark matter (SIDM) is an alternative to collisionless cold dark matter (CDM) which offers a unique solution to the problems encountered with CDM on sub-galactic scales. Although previous SIDM simulations have mainly considered elastic collisions, theoretical considerations motivate the existence of multi-state dark matter where transitions from the excited to the ground state are exothermic. In this work, we consider a self-interacting, two-state dark matter model with inelastic collisions, implemented in the Arepo code. We find that energy injection from inelastic self-interactions reduces the central density of the MW halo in a shorter timescale relative to the elastic scale, resulting in a larger core size. Inelastic collisions also isotropize the orbits, resulting in an overall lower velocity anisotropy for the inelastic MW halo. In the inner halo, the inelastic SIDM case (minor-to-major axis ratio $s \equiv c/a \approx 0.65$) is more spherical than the CDM ($s \approx 0.4$), but less spherical than the elastic SIDM case ($s \approx 0.75$). The speed distribution $f(v)$ of dark matter particles at the location of the Sun in the inelastic SIDM model shows a significant departure from the CDM model, with $f(v)$ falling more steeply at high speeds. In addition, the velocity kicks imparted during inelastic collisions produce unbound high-speed particles with velocities up to 500 km s$^{-1}$ throughout the halo. This implies that inelastic SIDM can potentially leave distinct signatures in direct detection experiments, relative to elastic SIDM and CDM.