The British Library's main site is on the Euston Road, though they also have a newspaper archive in Colindale.
Although you need a readers pass to access the collections held at the library, the building itself is open to the public, and there are generally a couple of exhibitions happenning. In addition there is a restauant and a coffee bar inside.
Most of the library, including the cofee bar and courtyard has wireless internet access (pay as you go, and seriously overpay, at that) - see their page about it.
As an aside, they have a bizarre domain name!
You think that's weird! You should see the British Museum's domain name. :)