I think it would depend on your budget. Can you afford to stay in the West
End at one of the great hotels. If so, I'd head for Brown's, my favorite.
If not I would stay in a area on the tube. There is a YWCA near Tottencourt
road and a Youth Hostel across from the British Library near Kings Cross
station. I've never stayed in either but like the locations.
You don't really want a car in London. There is lots of public transportation from bus, the Underground to the famous London taxi. I would take a 1/2 day bus tour that you can pick up in several places which will take you all over for brief visits and then if something really interests you, I would go back. It use to be that the British Museum was open on Sunday afternoons, but I'm not sure if it still is. London Tower is wonderful for wandering around. Leave this area by a river boat and take a ride up to Westminster and see London from water. Personally I love the Tate Museum and also try to get there for lunch which is expensive but a good experience. The Victoria and Albert is also a favorite museum for me. Friday night and Saturday night, I would go to a play. I love afternoon tea and so would head for Brown's or the Hyde Park Hotel for this activity. Personally I don't think Harrods is worth the time anymore. There are other great places to shop if you are interested in that. If you love books, Foyle's on Charring Cross road is great, at least five floors of books. If you like tennis, the tennis at Wimbledon is very interesting but it is a half hour at least from the center of London and a bit of a walk from the tube station, so your time gets eaten up getting there and back.
There is so much more, but time is fleeting and you have to decide what you like and head for that. Beth Sloan