I have always loved using the Paris Museum Pass while there. Is there a comparable one for London?
There are several museums and galleries that have Free entry: the British Museum, the National Gallery, the Natural History Museum, the Science Museum, Imperial War Museum, Victoria & Albert, the Tate Gallery. They do charge for special exhibits. We have used the National Trust UK membership pass to see several sites throughout England. They used to have an Overseas Visitor Pass, which you should inquire about for short stays. Others can advise on more passes.
Kathleen's provided a very good summary - you can keep yourself busy for at least a week with the large free museums in London. The smaller ones charge, but I can think of any passes that will cover them.
Apart from the National Trust, there's also English Heritage which is responsible for numerous historic sites, and offers both membership (which makes sense if you visit frequently) and an Overseas Visitors Pass.
and Historic Royal Palaces which, in London, has Tower of London (yes, it was a royal palace), Hampton Court Palace, Kensington Palace, Kew Palace, and Banqueting House (in Whitehall, where Charles I lost his head).
There is the London Pass but it is very expensive.
This comes as either an Explorer pass which allows you to visit a specified number of attractions over 60 days from first using it or a pass for a set number of days. Details here.
You would need to identify places you want to visit and then do the sums to see if it is worth your while to buy.