It's been 4.5 years since I was last in Paris and 7 since London. I'm happy to be in either for about a week in February. The main reason I haven't been is that rooms are so expensive in good weather (more so since I must have AC if there's a possibility of hot weather). I can bear cold and rain if it means being in either and can find lots of indoor things to enjoy in either city.
Budget is really just for a room in a good location. My basic requirements are clean, elevator or max. 3 flights of stairs, ensuite bathroom. On my visits to London I've stayed within walking distance of the British Museum (always spend hours there) and near a tube station, but I'm open to other locations, I'm just not familiar with them. In Paris I've usually stayed in the Marais, but I would consider close to Bastille in the 11th or the 5th or 6th, not too far south of the Seine.
The lowest rate for the place I love, the Morgan is now about £125/night. If I could find something in either place for around €80/night I'd take it and book a trip tomorrow.