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...
56 Crampton St
London SE17 3AE