cutting off the food source for the mites is more likely what got rid of them.
Neem only kills with a smothering agent (oil). Once it dries, it is no longer effective for the new hatchlings that are surely present.
Avid, for example is a contact killer as well as a short term (2-3 weeks) larvae killer, as it is translaminar, and resides within the upper layers of plant tissue. Any bug that chews the plant within that time will die as well, thus breaking the life cycle/chain of the mites.
Mites are as tenacious as they are damaging.