My landlady asked me not to move out in winter, but I want to leave in January. Is that okay?

I rented an apartment 10 years ago from an older woman. Initially I signed a one-year lease, but we went on a month-to-month basis after that. She did say when we went to month-to-month that she didn’t want me to leave in a hard-to-rent season, like fall or winter (in Chicago). She has been a decent enough landlord, though I deal mostly with her daughter and we don’t have much of a relationship.

I want to move out of my apartment this January. Do I have to pay rent through spring? Or can I rely on the legality of my agreement and disregard my landlord’s preferences?

— Mover