Unless you have driven overseas before, I really recommend that you not rent a car in London. Just trying to get out of the greater London area by car is nerve-wracking. If you have your heart set on a car, consider taking the train to Bath (about an hour from Pad and a nice trip) walk around to the major sites, have lunch (figure about 3-4 hours) and then pick up your rental car. From there, explore the beautiful country around Bath, which includes the Cotswolds. You would even be in fairly short driving distance from a couple of castles.
But you really can do a very nice two weeks in the UK with London as your base and no driving. London itself is wonderful, and with such a group as yours, you may only be able to fit in one major attraction per day once you add in travel and meal breaks. The kids would probably enjoy the Tower of London w/tour, the London Eye, the Science Museum and maybe the British Museum (it will depend on your kids). I know there have been lots of postings over the years re. kid friendly UK stuff.
As for day trips from London by train, there are so many to choose from and there are some really affordable options for train travel.