How long does it take to memorize a sonata?

The range of ability for memorizing varies hugely. I know people who can read through a score of a classical society, play it a couple of times, and they have got it. I know others that don't seem to be able to learn a large scale work in its entity no matter what they do. By the time they start work on the third movement they have forgotten the first. Your two to three months seems about average - but of course something you re-study several times over many years comes out better in the end.