Hi Karen
I lost weight through my low carb diet, I weighed 23st and did not exercise apart from walking, I built up on that though, gradually doing more, if you wanted to do something else though then you could always sneak into a gym, they are very discreet in there and are used to people that weigh more than they should, and they could give you advice on which exercises would be safe for you to do and you could possibly join a group of other people/women with very similar problems to yours.
You would get encouragement and guidance there.
If you are too shy to do this, then see if your doctor can refer you to physio and you will be given some exercises to do, you have to be careful with a back problem so that you dont make it worse, your spine must always be supported, a motorised treadmill is expensive and then you might not even use it very much, if I were you I would consider other options before spending so much money.
Also did you know that fidgeting around, making a 'fist' with your toes and relaxing, flexing your ankles and relaxing them, arm movements, etc maybe with some weights or something like cans of beans or small bottles filled with water, all done while you are sitting down comfortably with your back supported at all times can also be really beneficial and keep circulation going and use up some calories? Get a ball and while you are sitting down squeeze it between your knees and let go, and so on, loads of exercises that you can do either sitting down or even lying down.
All the best
Karen