We present a study of wind-driven nonlinear interfacial gravity waves using numerical simulations in two dimensions. We consider a case relevant to mixing phenomenon in astrophysical events such as novae in which the density ratio is approximately 1:10. Our physical setup follows the proposed mechanism of Miles [J. Fluid Mech. 3, 185 (1957)] for the amplification of such waves. Our results show good agreement with linear predictions for the growth of the waves. We explore how the wind strength affects the wave dynamics and the resulting mixing in the nonlinear stage. We identify two regimes of mixing, namely, the overturning and the cusp-breaking regimes. The former occurs when the wind is strong enough to overcome the gravitational potential barrier and overturn the wave. This result is in agreement with the common notion of turbulent mixing in which density gradients are increased to diffusion scales by the stretching of a series of vortices. In the latter case, mixing is the result of cusp instabilities. Although the wind is not strong enough to overturn the wave in this case, it can drive the wave up to a maximum amplitude where a singular structure at the cusp of the wave forms. Such structures are subject to various instabilities near the cusp that result in breaking the cusp. Mixing then results from these secondary instabilities and the spray-like structures that appear as a consequence of the breaking.