Often it can be hard to think of non-traditional/non-tourist activities to do in London, so I have complied a list of fun and different things to do in London, whilst also including some of the more traditional, tourist attractions.
London Activities:
Day trip to Brighton (check out my blog post on this)
Day trip to Oxford (check out my blog post on this)
Luxury cinemas (Electric Cinema, Curzon at Mondrian Hotel, Everyman Belsize Park, Olympia Cinema Barnes)
Shopping (Oxford Street, Sloane Square/Kings Road, Covent Garden, Canary Wharf, Westfield)
Afternoon Tea (Check out my Ultimate Brunch List post for the list of afternoon tea spots)
Food markets (Check out my Ultimate Brunch List post for the list of markets)
Bottomless Brunch (Check out my Ultimate Brunch List post for the list of bottomless brunch places)
Tourist Activities:
London Eye
Tower of London
Buckingham Palace
Kensington Palace
Hyde Park
Big Ben
Covent Garden
The Shard
Camden Town
Westminster Abbey
Hampton court
The Churchill War Rooms. They were used or recreated in the movie the Darkest Hour.

Galleries and Museums
Saatchi Gallery
Imperial War Museum
Victoria and Albert museum
The British Museum
The Tate Modern
The Portrait Gallery