We consider a rapidly rotating two-component Bose-Einstein condensate with short-range -wave interactions as well as dipolar coupling. We calculate the phase diagram of vortex lattice structures as a function of the intercomponent -wave interaction and the strength of the dipolar interaction. We find that the long-range interactions cause new vortex lattice structures to be stable and lead to a richer phase diagram. Our results reduce to the previously found lattice structures for short-range interactions and single-component dipolar gases in the corresponding limits.