I would like to add some comments. I agree with the above totally.
I understand, that on a clear day, you must do the London Eye. It looked boring to us, but my London friends say it is a marvelous experience.
Doug and I took a day trip to Bath. It is wonderful.
Other nice day trips are to Leeds Castle (Kent) and to Stratford/Cotwalds.
You can see Buckingham Palace. Call ahead for tickets.
The London bus tours are excellent and a good value..they include a boat trip.
Twofers for the theater are very doable. See above.
Airport to London. We took a cab and this past weekend, the Heathrow Express. It runs every 15 minutes and only takes 15 minutes to Paddington Station. It is 22 pounds round trip. Don't buy first class..it's a waste of money and is double. Just follow signs to Heathrow Express and buy the EXPRESS ticket from the machine with a credit card. Once at Paddington Station go to Track 1 and under the clock and you can pick up a cab from there. VERY efficient and just as good as a direct cab from the airport...faster, in fact.
Nobu at the Four Seasons Hotel is great.
John Frieda hair salons do a nice job of cut and color, if you need your hair done. They are fairly priced versus US.
I understand that the Jack the Ripper walking tour is wonderful.