This article concerns an extension of the topological sensitivity (TS) concept for 2D potential problems involving insulated cracks, whereby a misfit functional $J$ is expanded in powers of the characteristic size $a$ of a crack. Going beyond the standard TS, which evaluates (in the present context) the leading $O(a^{2})$ approximation of $J$, the higher-order TS established here for a small crack of arbitrarily given location and shape embedded in a 2-D region of arbitrary shape and conductivity yields the $O(a^{4})$ approximation of $J$. Simpler and more explicit versions of this formulation are obtained for a centrally-symmetric crack and a straight crack. A simple approximate global procedure for crack identification, based on minimizing the $O(a^{4})$ expansion of $J$ over a dense search grid, is proposed and demonstrated on a synthetic numerical example. BIE formulations are prominently used in both the mathematical treatment leading to the $O(a^{4})$ approximation of $J$ and the subsequent numerical experiments.