The problem is that your assumption that one can know the velocity with infinite precision is inconsistent with the practical requirement that you always get some position information. So the resolution is, the assumption is untrue. So why do we talk about "eigenstates" of momentum measurements, as if you could really get an infinitely precisely determined momentum? We do because physics is built around idealizations, that are not exactly realizable in practice but serve the purpose nevertheless. One should never take a "plane wave" solution too literally, all real wave functions are "wave packets", and do not have a perfectly precisely determined momentum-- and for just the reason you are asking about.