Understanding the possible structure of singularity models is a key step in many applications of Ricci flow. By work of Bamler it is known that the singular set of any non-collapsed limit of Ricci flows is of parabolic Minkowski codimension at least 4. It is an open problem whether this may be improved to locally finite codimension $4$ Minkowski content. In this talk we will discuss a new $\epsilon$-regularity theorem for the  K\"ahler Ricci flow and use it to derive sharp Minkowski estimates for the singular set under a finite energy condition. As an application, we give an alternative proof of Wang and Chen's estimate for singularity models of the Fano K\"ahler Ricci flow.