An intermediate-mass black hole (IMBH) may have a dark-matter (DM) minihalo around it and develop a spiky structure within less than a parsec from the IMBH. When a stellar mass object is captured by the minihalo, it eventually infalls into such an IMBH due to gravitational wave backreaction which in turn could be observed directly by future space-borne gravitational wave experiments such as eLISA and NGO. In this Letter, we show that the gravitational wave (GW) detectability strongly depends on the radial profile of the DM distribution. So if the GW is detected, the power index, that is, the DM density distribution, would be determined very accurately. The DM density distribution obtained would make it clear how the IMBH has evolved from a seed black hole and whether the IMBH has experienced major mergers in the past. Unlike the γ-ray observations of DM annihilation, GW is just sensitive to the radial profile of the DM distribution and even to noninteracting DM. Hence, the effect we demonstrate here can be used as a new and powerful probe into DM properties.