join the city libraries so you can use barbican and guildhall - barbican is the nicest public library i've been to and has all the sheet music, pianos etc. interesting stock of old crime novels and lotsa literature too. best in london. guildhall has the largest single-city collection (on london) in the world apparently
camden is also handy cos then you can use holborn library if you're in the west end, which is frequented by the notorious "milk-man", a guy who is always, always in there on a computer with a 4-pint bottle of milk, reading the daily mail website. reasonable stock
anyone can join the poetry library at the festival hall where you will find more interesting poetry than anyone could read in a lifetime,sometimes interesting displays etc, also my book is in the collection!!
you might not be able to join idk but i like UCL library as an ex-student there and definitely a nicer place to be than nearby senate house library though they may have more stock, certainly no shortage at UCL