Surfactants are used in a wide range of chemical and biological applications, and for pharmaceutical purposes are frequently employed to enhance the solubility of poorly water soluble drugs. In this study, all-atom molecular dynamics (MD) simulations and small-angle neutron scattering (SANS) experiments have been used to investigate the drug solubilisation capabilities of the micelles that result from 10 wt% aqueous solutions of the non-ionic surfactant, Triton X-100 (TX-100). Specifically, we have investigated the solubilisation of saturation amounts of the sodium salts of two nonsteroidal anti-inflammatory drugs: ibuprofen and indomethacin. We find that the ibuprofen-loaded micelles are more non-spherical than the indomethacin-loaded micelles which are in turn even more non-spherical than the TX-100 micelles that form in the absence of any drug. Our simulations show that the TX-100 micelles are able to solubilise twice as many indomethacin molecules as ibuprofen molecules, and the indomethacin molecules form larger aggregates in the core of the micelle than ibuprofen. These large indomethacin aggregates result in the destabilisation of the TX-100 micelle, which leads to an increase in the amount of water inside of the core of the micelle. These combined effects cause the eventual division of the indomethacin-loaded micelle into two daughter micelles. These results provide a mechanistic description of how drug interactions can affect the stability of the resulting nanoparticles.