This year and last year I went to London and there are so many things to see and do. I went on the London tour in 2005, it was my first Rick Steves tour and I was hooked. I just got back from London and can't wait to go back!
As everyone said, the Sir John Soane's musuem is amazing and should only take one to two hours. The Wallace collection is another wonderful museum to visit and is down the street from Selfridges ( there is an excellent cafe I went to this year in Selfridges for lunch) and of course tons of plays to see. Last year I saw Wicked and this year I saw Aladdin. I had tea with friends at Forum & Mason and that was special.
I totally loved the Tate Britian and the National Portrait Gallery! Of course there is the British Musuem but I haven't been there since 2005. My all time favorite is the Victoria & Albert Museum but that is huge so you will need to choose the galleries carefully to see what you really want to see. The National Gallery is another one to see.
The Tate Modern is interesting but I think once is enough for me to see it. Not my favorite museum.
St Paul's Catherdral is exceptional to see.
The London walks are wonderful but really not sure if you would want to do that in your free time as you will be doing walks with the tour. I did take a pub crawl this year and last year and they were a lot of fun. You may want to do that one evening. You go on the London walks pub crawl, just show up at the time usually around 7PM at a tube station and they take you on the walk of the area and to three pubs. I had a great time both times.
There are tons of pubs to visit and have a drink or a bite to eat. Let me know if you want any suggestions.
If you fall in love with London and want to return, Windsor Castle is a wonderful day trip where I saw the changing of the guard as is Hampton Court Palace with wonderful gardens. I have my heart set on going to Kew Gardens on my next visit.
If you decide to go back to London from Aug to Oct you can see the inside of Buckingham Palace. Of cours this is for future visits.
I also went to see Harrods and the food hall and had tea and pastry at their Laduree branch and went on every floor of Fortum and Mason to see that beautiful department store.
There is also walking across the Tower Bridge and you can go to Greenwich too.
Hope this helps.