Quantitative time-dependent laboratory measurements are made of the irreversible mixing caused by the development, saturation and turbulent breakdown of Kelvin-Helmholtz (KH) billows in an initially stable two-layer stratified shear flow of miscible fluids. The measurements are obtained using a light attenuation technique. Irreversible mixing is found to be strongly dependent on the life-cycle of the flow, in particular the characteristics of large-scale, quasi-two-dimensional billow merging events, which stir the flow and then trigger mixing during turbulent transition.