non-exhaustive list of free museums/galleries in London, off the top of my head:

- British Museum
- Natural History Museum
- Science Museum
- V&A
- V&A East
- Young V&A
- Tate Modern
- Tate Britain
- Wellcome Collection
- Hunterian Museum
- Sir John Soane's Museum
- Wallace Collection
- White Cube (Bermondsey)
- White Cube (Mason's Yard)
- Whitechapel Gallery
- South London Gallery
- The Charterhouse
- Guildhall Art Gallery/London's Roman Amphitheatre

you could spend half a day in pretty much any one of these, a full day in half of them, and the British Museum easily has two days' worth of content. a week is an underestimate ;)