I realise London is full of them but today I discovered...
THE OPEN MUSIC ARCHIVE
What it says on the tin but to expand: There is a lot of music in the public domain which has fallen out of copyright however a lot of it is tied to physical media (e.g. gramophone records) and locked away in archives or private collections which are not widely accessible. The Open Music Archive aims to digitise as much of this music as possible in order to free it from the constraints of a physical collection. It's open to anyone and they're trying to expand it all the time. Dope.
THE BFI MEDIATHEQUE
THOUSANDS of films, television programmes and archive footage of Britain you probably can't see anywhere else...just ring up or show up if you feel like it, book a HUGE screen, and enjoy.
I love London.