The population of Milky Way (MW) satellites contains the faintest known galaxies and thus provides essential insight into galaxy formation and dark matter microphysics. Here we combine a model of the galaxy-halo connection with newly derived observational selection functions based on searches for satellites in photometric surveys over nearly the entire high Galactic latitude sky. In particular, we use cosmological zoom-in simulations of MW-like halos that include realistic Large Magellanic Cloud (LMC) analogs to fit the position-dependent MW satellite luminosity function. We report decisive evidence for the statistical impact of the LMC on the MW satellite population due to an estimated 6 2 observed LMC-associated satellites, consistent with the number of LMC satellites inferred from Gaia proper-motion measurements, confirming the predictions of cold dark matter models for the existence of satellites within satellite halos. Moreover, we infer that the LMC fell into the MW within the last 2 Gyr at high confidence. Based on our detailed full-sky modeling, we find that the faintest observed satellites inhabit halos with peak virial masses below at 95% confidence, and we place the first robust constraints on the fraction of halos that host galaxies in this regime. We predict that the faintest detectable satellites occupy halos with peak virial masses above, highlighting the potential for powerful galaxy formation and dark matter constraints from future dwarf galaxy searches.