We have observed 28 heteronuclear Feshbach resonances in ten spin combinations of the hyperfine ground states of a $^{40}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ mixture. The measurements were performed by observing the loss rates from an atomic mixture at magnetic fields between $0$ and $700\phantom{\rule{0.3em}{0ex}}\mathrm{G}$. These data were used to significantly refine an interatomic potential derived from molecular spectroscopy, yielding a highly consistent model of the $^{40}\mathrm{K}\text{\ensuremath{-}}^{87}\mathrm{Rb}$ interaction. Thus, the measured resonances can be assigned to the corresponding molecular states. In addition, this potential allows for an accurate calculation of the energy differences between highly excited levels and the rovibrational ground level. This information is of particular relevance for the formation of deeply bound heteronuclear molecules. Finally, the model is used to predict Feshbach resonances in mixtures of $^{87}\mathrm{Rb}$ combined with $^{39}\mathrm{K}$ or $^{41}\mathrm{K}$.