In the context of geometric evolution equations a common method is to use a level set function to model the evolution of surfaces in the three dimensional Euclidean space. While $C^0$ is a natural regularity in the context of viscosity solutions in special cases higher regularity is available and even necessary in order to detect level sets from the level set function in the geometrically rather non-pathological case of smooth surfaces as level sets... Correspondingly, for a finite element scheme approximating the level set equation one would like to also have error estimates which usually come naturally in Sobolev norms at least in $H^2$ which then imply via embedding theorems an error estimate in $C^1$ and hence give meaningful information about the approximation error of the level set surface itself. This paper proves error estimates for $H^2$ conforming finite elements for equations which model the flow of surfaces by different powers of the mean curvature (this includes mean curvature flow). The scheme is based on a known regularization procedure and produces different kinds of errors, like regularization error, FE approximation error for the regularized problems and full error. While in the literature and own previous work different aspects of the aforementioned error types are treated, here, we solely and for the first time focus on the FE approximation error in the $H^{2,\mu}$ norm for the regularized equation inclusively the aspect of working out all dependencies from the regularization parameter explicitly. (read more)