The forward enstrophy cascade in two-dimensional quantum turbulence in a superfluid film connected to a thermal bath is investigated using a Fokker-Planck equation based on Kosterlitz-Thouless renormalization.  The steady-state cascade is formed by injecting vortex pairs with the same large initial separation (the stirring scale) at a constant rate.  They diffuse with a constant flux to smaller scales under the action of frictional forces, finally reaching the core size separation, where they annihilate at the same rate they are injected and the energy is removed by the thermal bath.   The energy spectrum varies as k^{-3}, similar to the spectrum known for 2D classical enstrophy cascade.  The dynamics of the cascade can also be studied in switching on and off the injection, and in the decay from arbitrary initial vortex-pair  distributions.  For the case of an initial distribution sharply peaked at a particular vortex separation, it takes about 3-4 eddy turnover times for the system to evolve to the decaying k^{-3} cascade, in agreement with recent computer simulations [1] of the enstrophy cascade.  These insights into the nature of the cascade also allow a better understanding of the phase-ordering process of temperature-quenched 2D superfluids.  The decay of the vorticity present in the high-temperature phase in fact proceeds via a constant-flux enstrophy cascade.  This connection with turbulence may be a fundamental characteristic of phase-ordering in general.