Last summer, my hubs and I took our three kids (15, 13, 8) to London for nine days. We hit a lot of spots, including:

Museum of London
Tower of London
Hampton Court
Westminster
St. Martin-in-the-Fields
saw War Horse
did a Ghost Walk
London Eye
went to Stonehenge
spent time at Portobello Road market
Kensington Gardens (palace was being renovated)
Churchill War Rooms
walked a lot -- down Piccadilly, around St. Paul's (although we didn't tour)

This time, we're going to the British Museum via London Walks. The hubs didn't get to go to Stonehenge becaue he was working, so he wants to go, and I want to spend more time at the Salibury Cathedral. We want to see Sweeney Todd. I'm going to Fortnum and Mason because I want to see the store.

What other suggestions do you have? I know we missed lots. I want to go to Spitalfields and the Borough Market. Hubs has to work one day in Parliament, and I'm going as a note-taker (already OK'd. Very excited about it.)

We have eclectic tastes and love things off the beaten path, so suggest away! What's your very favorite London spot??

We're leaving in a week.

My favorite London spot is Kenwood House in Hampstead Heath. but right now it is closed. When are you going to London?

oooooops.

Just saw you are leaving in a week.

My second favorite spot is the Globe theatre. They're still performing the Taming of the Shrew ( I think.)

We're leaving in a week.

Ooh, I like that one! The heroine is Kate -- that's my name, too!

The whole South Bank area (around the Globe, Eye, etc.) and the Tate I found interesting and worth doing. (Last week.)

Forgot the Tate! Thanks for reminding me!

Buckingham Palace should be open which I enjoyed.
We particularly liked the Treasures of the British Library and the Victoria & Albert Museum.
If you have time, a day trip to Bath would be fun.

Without children, enjoy drinks in a bar or pub after your dinner, or club/disco if it's your scene (some have over 25's nights). You can spend hours in your favourite museums or art galleries, or enjoy shopping at leisure. What about an afternoon tea at the Orangery in Kensington Palace, or a high-end restaurant for dinner? In the recession, getting a table has become much easier. Or a dinner cruise on the Thames.

Have done the London Walks British Museum tour, with guide Tom. Was a terrific introduction, wouldn't have known where to start without it. Guess you are combining Stonehenge and Salisbury or I would have suggested Winchester instead of Salisbury - on the outside Salisbury is more impressive but I think Winchester is much more interesting inside (there are also other things to do in Winchester - the round table in the Castle and Winchester college.)
One thing I have in my diary for what's coming up next week is the start of a pre-raphaelite exhibition at Tate Britain - very much my interest, but, of course, may not be of interest to you.
You have discovered London Walks already - if I have spare time I often see what they have on offer and turn up, some of the more unusual walks are very interesting.

Mine are Leighten House - Open again after getting refurbished. It is the home of Frederic, Lord Leighton (pre-raphaelite artist) and the Arab Hall is fabulous.

John Soane Museum - He was an architect and this is his home/workspace and is stuffed with all sorts of architectural bits and pieces. But my favorite is his picture collection, it contains the original oil paintings of Hogarth's "A Rakes Progress." Its also fun to watch the attendant move the walls around to show off the pictures (you have to see it to know what I mean).

And one more fun thing, where New Bond St. and Old Bond St. meet, there is a bronze bench with a place for someone to sit to have their picture taken with Churchill and Roosevelt.

Awesome suggestions! Thanks!

The V&A Museum. I especially enjoyed the furniture gallery.

A quick thought. Dont know if you'll still be in London then, but this year's Open House London is on the 22nd/23rd of September. There is a website with details of what is open if you google Open House London 2012 you should find the website.
You can get into buildings you can't normally access - some pretty quirky.