I've had to deal with this so many times!!! Very first thing on the list is to get a wet vac. You can rent them, but I'm guessing there will be a neighbor somewhere that has one. Ask around. It will help TONS. If you can't find one, call your landlord. Tell him it's from the water heater (which is also his responsibility) and he needs to take care of it before it gets completely moldy and has to be replaced. You've been doing everything you can and it's his turn. But really he needs to be notified anyway.
The smell is because of the carpet padding. It gets gross after a few years and it will always smell, especially if the water isn't taken care of immediately.
After you get all the water up and it's mostly dry, buy a thing of baking soda and sprinkle it heavily on the affected area. Let it sit till you can't smell anything and then vacuum it up. You can repeat the process a few times if you think it's needed. Also, Febreze for carpet (or for furniture, at least) will help with any smell left over.