Lambeth Palace Library

LPL is the historic library of the Archbishop of Canterbury, and the first port of call for any research of the Church of England, as well as wider ecclesiastical history.
It is more of an archive than a library as it operates with strict archive regulations. The reading room is spacious, light, airy, and looks out over a pretty courtyard. I think it is quite easy to get a reading card, but they do have a few requirements, so check their website and get the stuff together before you visit because they will send you away unless you’ve got the right ID documents. The library was founded in 1610, so they do have to be strict about a lot of the older and rarer documents.
The library is not unwelcoming though. You can leave your sandwiches in their fridge, and help yourself to tea and coffee for a few pence each.