This is my old "random things to do" list (with a few no longer valid things pruned):

Wildlife Photographer of the Year exhibition at the Natural History Museum
Mousetrap (Theatre) - been running without interruption for 52 years
Lighthouse Restaurant in St. Katherine's Docks
Alessi coffee and tea towers in Lincoln's Inn
Sotheby's have a wine tasting course on
Smithfield Meat Market, followed by brunch at Smiths of Smithfield
Turkish and Russian Baths at York Hall
London Jazz Week is on that week, loooads of stuff on.
Hampstead village then walk on Heath, ending up in Spaniards Inn pub.
DLR out to Greenwich, through the market, up to the dateline and Royal Observatory, then mainline train back to central.
Jack the Ripper walk around Aldgate
London Duck Tours It's crazy, embarrassing, but lots of fun!  www.londonducktours.com
Visit to Parliament, including going inside Commons or House of Lords…
Chai Bazaar (Indian tea bar),
Yauatcha (Chinese tea bar)
Markets - Billingsgate Fish Market, Smithfields or Borough
The largest Hindu temple outside India, in Neasden, and it's just after Diwali this weekend. Shri Swaminarayan Mandhir
One of the many Comedy Clubs
MASH or Freedom Bars - they brew their own beer
Belgos - 300 beers plus mussels
Garlic and Shots restaurant in Soho - everything garlicy, even beer and puddings!
The "Monument" - climb up it and see views of city from top
Canal - take canal boat from Camden to Little Venice or vice versa. Little Venice is very pretty.
Riverside pubs at Richmond, and walk in Richmond Park.
Pineapple Dance Club - take a lesson in some form of dancing.. salsa or anything
Manor Park rock climbing club.. beginners lesson?
The Fat Duck http://www.fatduck.co.uk/  Sublime foodie stuff.. Three michelin stars. Out in Bray in Berkshire. Holy!
Locanda Locatelli http://www.locandalocatelli.com/ Best of Italian in London at the moment..
London Transport Museum
Shakespeare's Globe
Royal Botanic Gardens Kew
London Walks. Ripper Walk, The Westminster Nobody Knows sounds interesting, or Ghosts of the West End
Pizza Express Soho Jazz
Microbrewery in Greenwich
Hampton Court Palace
Strangers Gallery in Parliament
Alton Towers trip
British Grand Prix
Bramah Tea and Coffee Museum
Sing-A-Long-A Sound of Music at Prince Charles cinema - even more fun dressed up as Nun or Nazi!
Weird restaurants like Archipelago - serving scorpions etc
Auction houses
Millennium footbridge and Tate Modern
Kyoto Garden in Holland Park
Disco Roast in Shoreditch
RollerSkating in Hyde Park
Dennis Severs’ House Tour
Folk and Blues at the Harrison, King's Cross
Kew Bridge Steam Museum with London's only working Steam Railway
Urban Golf in Soho - high tech golf simulation
East London Art Walk - guided around some of the newer galleries
Speakers Corner
Farmers' Markets
Bike Polo in Brick Lane - bring your bicycle, mallets are provided
Columbia Road Flower Market, plus oysters in the alley that connects Ezra Street and Shipton Street next to Jones’s equally inviting dairy store. 
Sunday Knit Roast
Free Brazilian dance Forro lessons at Guanabara on Drury Lane
Boules in Cleaver Square in Kennington

Apsley House & Wellington Arch 

Bank of England Museum

The British Dental Association Museum 

Brunel Engine House

Cartoon Museum

Cabinet War Rooms and Churchill Museum

The Museum of Brands and Packaging

Horniman Museum

Freud Museum

Petrie Museum - Egyptian stuff

Garden Museum and Cafe

Crystal Palace Museum

Guildhall Clockmakers' Museum

Hunterian Museum

Kirkaldy Testing Museum

Leighton House

Library and Museum of Freemasonry

London Canal Museum

London Fire Brigade Museum

London Sewing Machine Museum 

MCC Museum

Museum and Library of the Order of St John 

Museum of Immigration and Diversity

Museum of London

Museum of Rugby

New London Architecture

Royal Academy of Music

Sherlock Holmes Museum

Sir John Soane’s Museum

Smythson Stationery Museum 

Wellcome Institute

The Old Bailey for a trial

Wallace Collection

Hackney City Farm

Mudchute Farm

Tennis for Free

Harrods

British Library

Pure Groove record shop, EC1

Greenwich Meridien etc

Southwark Cathedral Audio Tour

Roundhouse in Camden

Wimbledon Stadium Greyhound Racing

Prince Charles Cinema

Windmill in Brixton for live music

New music and poetry on a boat for just £6 (£4 concs) at Yacht Club, Victoria Embankmemt

Vampire Walk in Highgate