We present a comprehensive investigation of the Berezinskii-Kosterlitz-Thouless transition in ultrathin strongly disordered NbN films. Measurements of resistance, current-voltage characteristics, and kinetic inductance on the very same device reveal a consistent picture of a sharp unbinding transition of vortex-antivortex pairs that fit standard renormalization group theory without extra assumptions in terms of inhomogeneity. Our experiments demonstrate that the previously observed broadening of the transition is not an intrinsic feature of strongly disordered superconductors and provide a clean starting point for the study of dynamical effects at the Berezinskii-Kosterlitz-Thouless transition.