Sleeping on the floor is not going to do you any good - that's a myth. Maybe you do need a new mattress, but a clapped-out bed wouldn't do most people this much damage - your doctor is just trying to get rid of you. In my experience, GPs are terrible with back pain and don't understand it one bit.
If it's come on gradually without any major injury, I would suggest your pain has something to do with posture or perhaps an inherited condition. The only thing that's helped with my back problems is visiting an osteopath - if you can find a good one, they're well worth it. Unlike a chiropractor, an osteopath will look at your whole body and address any issue that crops up, as your back problem could have something to do with your balance, the shape of your spine, or even an old neck or leg injury.
The last time I asked, GPs weren't allowed to refer you to an osteopath, so you'll probably have to find one yourself. The first session will cost more than subsequent ones, as it will include a long consultation about what's wrong with you.
I wouldn't bother with a physio if I were you, as they've never done me any good, and neither have chiropractors (in fact, a very evil chiropractor actually made me worse and I was in bed for a month). But lots of people swear by them, so I suppose you don't know what's going to work.
So, in conclusion, sleep on a bed and find an osteopath. And I find that ibuprofen does the trick when I'm in a state.