What sort of books are they? I know that in the Knowledge & Information Centre on the ground floor of St Thomas' they have a shelf where books are sold for 1 pound each. I'm told they have a trolley taking books round the wards. You could try calling them and asking them if they'd be interested.
56a Infoshop behind Walworth Rd, close to Strata, is a local community space that runs a free book exchange. Basically people bring a book they have finished and select one they want to take. It's a book swap point really. They take donations of books to keep the exchange supplied...
56a Infoshop
56 Crampton St
Pullens Estate
London SE17 3AE
I usually take books that are in good condition to the Oxfam Bookshop on Stutton Ground in Victoria (just off Victoria St). They welcome donations and it's actually a really nice book store.
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