I personally prefer staying in flats as opposed to hotels now that we have kids — it gives us more space and freedom. I recently stumbled over this website called VRBO (Vacation Rentals By Owner) whilst looked for a short term rental flat for a friend in Paris and I think it is one of the best websites I have seen. The big advantage is this: the site puts you in direct contact with the owner of the flat so that you don’t need to deal with an agency. This saves you a lot of money in agency fees. It also means that you deal directly with someone who has invested themselves in their flat, mostly meaning you get a better, more personal service. There are a lot of family-friendly flats on the site in central locations, and most of the owners I have been in contact with spoke fluent English.
I have no idea if VRBO works well for other cities, but for finding short term accommodation in Paris, it has been great.