And for the defence as someone who lives in London and loves it:
Getting Around: You can buy daily travel cards that are valid on buses, the tube and the rail network within London. Probably your best bet there.
Stuff to do: Most of the main museums are free, British Museum, Natural History Museum, Science Museum, V&A. Same goes for the Tate Galleries and the National Gallery. My favourite museum is The John Soane on Lincoln's Inn Field. Places like Covent Garden, Lamb's Conduit Street and Spitalfields are good for general boutiquey shopping (clothes etc.). Most of the guitar shops are on Denmark Street. I like Vintage and Rare Guitars, and Wunjo. Angel Music's supposed to be good too. Areas like The South Bank around the Royal Festival Hall are good for a wander.
Does that help at all?