The satellites of the Milky Way have been often used as a test-bed for cosmological models, for instance because they are strongly dark matter dominated objects. However it is on these scales where the currently favored LCDM model faces most challenges. Recently, Boylan-Kolchin et al (2010) highlighted another potential discrepancy between observations of the local population of dwarf spheroidal galaxies (dSph) and the results of high resolution simulations of subhalos assembled in fully cosmological context. Firstly, simulations would seem to produce objects that are much more concentrated than observed, i.e. they would reach their peak circular velocity at a smaller radius than estimated for the dSphs. Secondly, the number of bright satellites having maximum circular velocities in excess of a given value would appear to over-estimated in LCDM simulations. In this work we present solutions to both problems by revisiting two basic assumptions tacitly made in previous studies, namely, the density profile followed by dark matter subhalos, and the mass of the Milky Way's dark matter halo.