Direct force measurements between negatively charged silica microparticles are carried out in suspensions of like-charged nanoparticles with atomic force microscopy (AFM). In agreement with previous studies, oscillatory force profiles are observed at larger separation distances. At smaller distances, however, soft and strongly repulsive forces are present. These forces are caused by double layer repulsion between the like-charged surfaces and can be quantitatively interpreted with the Poisson-Boltzmann (PB) model. However, the PB model must be adapted to a strongly asymmetric electrolyte to capture the nonexponential nature of these forces. Thereby, the nanoparticles are modeled as highly charged co-ions, while the counter ions are monovalent. This model permits extraction of the effective charge of the nanoparticles, which is well comparable to the one obtained from electrophoresis. The PB model also explains the presence of a particle-free layer close to the interface.