Museums are free, and there are loads in London: Victoria and Albert, Natural History Museum, British Museum, both Tates (Tate Britain and Tate Modern), Imperial War Museum, National Gallery, etc.
You can relax in Trafalgar Square (mind the pigeons), Leicester Square, take a bus and go to Greenwich, to see the Cutty Sark (if they let you), Maritime Museum, the Greenwich Meridian. Covent Garden, they always have interesting activities. Have a walk in Westminster and see Big Ben and houses of Parliament, St. James Park (to have a picnic and see the ducks, swans, etc), a tour in Embankment and if you walk along the river you can see the London Tower, the Tower Bridge, Gherkin. You can spend an afternoon in Hyde Park, Green Park. Have a go in Regent Street, Carnaby, go to Liberty's just to see the building and the things they are selling. You can go to Harrods in Knightsbridge, only to look around.
The cruise in the river Thames is fun, but as I recall it's £11 per person to go to the barrier, stay a while in Greenwich and then back to the pier.
Without much money you can't visit the attractions in London, like the zoo, the London Aquarium, Tower of London, London eye, etc.
Enjoy your stay in London, it is is a lovely town.