Direct detection of dark matter on the Earth depends crucially on its density and its velocity distribution on a milliparsec scale. Conventional N-body simulations are unable to access this scale, making the development of other approaches necessary. In this paper, we apply the method developed by Fantin, Merrifield and Green in 2008 to a cosmologically based merger tree, transforming it into a useful instrument to reproduce and analyse the merger history of a Milky Way-like system. The aim of the model is to investigate the implications of any ultrafine structure for the current and next generation of directional dark matter detectors. We find that the velocity distribution of a Milky Way-like galaxy is almost smooth, due to the overlap of many streams of particles generated by multiple mergers. Only the merger of a 1010 M⊙ subhalo can generate significant features in the ultralocal velocity distribution, detectable at the resolution attainable by current experiments.