Please do not take these dogs to Battersea. If you do they can (and probably will) put them to sleep immediately. There are much cuter, younger dogs that need homes and people will almost always choose these first. Older dogs, especially Staffs have little chance of finding a home.
I think OP you need to ask yourself if you are prepared to take your dogs to the vets and have them put to sleep, because if you don't keep them this is almost certainly what will happen to the Staff at least. I appreciate that is harsh but there just aren't the homes available out there, hence around 25,000 dogs being put to sleep each year in the UK. Too many dogs, even lovely, healthy ones and not enough homes. 
I do appreciate your situation is a difficult one, do you think maybe if you explain to the landlord that your dogs are elderly and may not even live much longer they will reconsider? Can you afford to cut costs somehow so that you could afford to pay a little extra rent? (not sure how housing benefit works, but would that help with this?) Do you think if you explained to your landlord that rescue places are few and far between and your only real option in the short term would be to have them put to sleep he would let you keep them until a new home of rescue place became available? (Maybe a rescue could give you a letter saying that their waiting time was approx xx months and your dogs were on the waiting list?)
Some rescues will also add your dog to their website whilst they stay with you, so they don't need to be kennelled, maybe this is an option you could consider?
Whatever, please do not consider rehoming yourself without using a rescue, especially in London. People in dog fighting circles find dogs this way and are extremely good at convincing you they are a good home, including bringing children to see the dog etc.