There is mention of Lambeth Hospital (as it came to be known) on the National Archives site - but it says that all records are with London Metropolitan Archives.
http://discovery.nationalarchives.gov.uk/details/r/4a5e1172-d9b7-4231-9108-3d0abc01d177London Metrpolitan Archives says:
Over 80 hospitals, many now closed, have deposited their records in London Metropolitan Archives (LMA). The extent of our holdings for each hospital ranges from one volume or document to hundreds of feet of archives.
Some hospital records have not yet been catalogued and can only be made available for consultation by prior appointment at least a week in advance.
Access to Patients Records
Hospital records are subject to a period of restricted access in order to protect the confidentiality of living individuals. It may therefore not be possible for you to consult all the records yourself. We are able to undertake research on behalf of an individual to provide information from hospital records but we do have to ensure that the information is only being released to either the individual concerned or in the case of a third party request that the patient concerned is deceased. We do charge for undertaking such searches within our records.
We suggest that in the first instance you contact the enquiry team who will be able to advise you on how to proceed with your particular enquiry.