Smithfield Market at dawn. Everything finishes by 8.00 am. Have breakfast and a pint in one of the pubs and watch the meat porters going about their business. Great fun.
Borough Market, Londons oldest food market. Bermondsey Market for antiques.
Spitalfields for great art and craft ships and brilliant restaurants. Has one of the best Tapas restaurants in London.
http://www.londonmarkets.co.uk/
Go for a picnic and concert at Kenwood House. Last year they had Diana Krall, Van Morrison and Katherine Jenkins.
Walk beside the Thames from Waterloo through to Rotherhithe stopping at each pub on the way.
Have a drink in Vertigo 42, the bar at the top of the Nat West Tower in the City. http://www.vertigo42.co.uk/ Not cheap but worth it for the views.
As for theatres and culture, try the Almeida
http://www.almeida.co.uk/
The LondonBridge Festival
http://www.londonbridgefestival.com/showsandevents/
Time Out is the best guide to London's cultural diversity.
http://www.timeout.com/london/
Museums. Handle a gold bar at the Bank museum or see Dr Barnardo's Ragged Day School museum.
Cycling routes
http://www.timeout.com/london/aroundtown/features/8359/London_cycling_routes.html
Walking tours
http://www.walks.com/
Rules - as English as an English restaurant can get. Treat yourself.
Manzes Pie and Mash. Very East End.
Fortnum & Mason for luxury food from all around the British Empire (as was) to take back with you.
How long are you in London for?