I'll also endorse making London your base and making just enough day trips to allow you to see at least a few of the daytime London sites like the British Museum, the Victoria and Albert Museum, the National Gallery and Trafalgar Square, and perhaps the Churchill War Rooms. If there is then a day or two for day trips, there are many choices and, since the natives mostly speak some variety of English, most day trips can be happily done without a guide. In this fashion we've traveled to Canterbury, Cambridge, Greenwich and Brighton and got a great trip out and return in a day each time,,with longer stays in Bath and York (but Bath works forva nice day trip as well).
Most recommended, as have others, if you have a few days to be based in London renting a flat is indeed much better than a hotel. We've happily used Utah-based London Connections three times, once in Knightsbridge and twice in Mayfair, most recently in September. In addition to having three or four evening meals in our flat (sometimes something as simple as a salad, sandwich from a Pret a Manger and a beer or two) after a full day's walking can hit just the right shoes-off spot. A particular delight for us has been the economical and most pleasurable self-made breakfasts with our own French Press coffee, an egg or two, and toast with genuine English orange marmalade and the morning London Times to share at our quiet breakfast table.