I'd reccommend joining www.streetcar.co.uk . I don't have anything to do with them but we used to hire from Enterprise and Streetcar is just so much easier...loads of cars in SE1 and they are 'open' 24 hours a day, 7 days a week! Best of all you can hire for any length of time form 1 hour to 1 week.
I am a member at streetcar as well and so far, I cannot complain. If you rent a car for 24 hours, up to 50 miles are free, after that you have to pay - I think it's 23p per mile- but petrol is free.
A friend recently recommended to rent a car at City airport, that seems to be the closest place from SE1 and if the Jubilee Line is not suspended, it is easy to get there.