A common situation in galactic and intergalactic gas involves cold dense gas in motion relative to hot diffuse gas. Kelvin-Helmholtz instability creates a turbulent mixing layer and populates the intermediate-temperature phase, which often cools rapidly. The energy lost to cooling is balanced by the advection of hot high enthalpy gas into the mixing layer, resulting in growth and acceleration of the cold phase. This process may play a major role in determining the interstellar medium and circumgalactic medium phase structure, and accelerating cold gas in galactic winds and cosmic filaments. Cooling in these mixing layers occurs in a thin corrugated sheet, which we argue has an area with fractal dimension D = 5/2 and a thickness that adjusts to match the hot phase mixing time to the cooling time. These cooling sheet properties form the basis of a new model for how the cooling rate and hot gas inflow velocity depend on the size L, cooling time of the mixed phase , relative velocity , and density contrast of the system. Entrainment is expected to be enhanced in environments with short , large , and large . Using a large suite of three-dimensional hydrodynamic simulations, we demonstrate that this fractal cooling layer model accurately captures the energetics and evolution of turbulent interfaces and can therefore be used as a foundation for understanding multiphase mixing with strong radiative cooling.