There is so much you can do...the majority of museums are free, id go to the natural living museum its amazing they have the dinosaurs and things, also a walk along south bank is amazing its so pretty so go on a good day.
Also just generally walking around Leicester square, maybe watch a premier, walk down to covent garden from there.
Visit Trafalgar square, 10 downing street, the cathedrals, big ben, Leicester square, hyde park, the palace, oxford circus- window shopping, bond street
Also a MUST is notting hill, the houses are amazing and on a weekend they have the portobello market which is Amazing just to look around- they sell really random antiques and gorgeous little things you wont find anywhere else, vintage clothes etc.... The houses in portobello are so cute also nice to have a look.
The flower market is also meant to be wonderful to see...
theres literally so much...id just say get off at a main tube stop and just stumble around soho or wherever and you will eventually pass lots of really cool places
lived in london as a very poor intern