Ah, London. So much to do, so much to see. The only thing we are short on is space. You are, of course welcome to study in study lounges in your halls of residence, in your kitchens, or the Bell Classroom in the Crofton (after classes are over), but here is a short list of other suggestions:
The Kensington and Chelsea Library just off Kensington High Street
Caffe Forum on Gloucester Road
Not so near:
University of London Union, this will require joining the union for the term, but it also entitles you to entry at all the events, and the ability to join clubs and societies. A great way to meet students from all University of London campuses.
Ozone Coffee Shop near Old Street
Photo courtesy of Lifehacker
Fix, in Shoreditch
Ziferblat, where everything is free except time. You can sip coffee or tea, make yourself a snack, and wash up after yourself for 3 pence a minute.