First & foremost, my thoughts, as were Walkin's that if it's only backwards she doesn't do, I'd get her checked out physically. A good bodyworker can often pick & fix problems not picked up by a vet. Given her previous life, I'd probably want to get that done before starting much training with her too.
Yes, that you've let her get away with stuff, 'mug you' for treats etc won't have helped. Don't blame her, it's just up to you to teach her better. She's just a horse, will do what works for her, or what she's allowed & will quit doing what doesn't work. So you need to think about the 'manners' you want her to learn & be consistent & effective in making them clear.
Be *effective* in whatever you ask. I like to use non-confrontational methods, and I find there's usually a reason that a horse doesn't understand & yield(respond softly & willingly) to pressure(you're not being clear enough, there's another horse or something fearful in the way... whatever), so be conscious about setting up the lesson for success - make it as clear & easy as you can for her to get it right, so she can practice being reinforced for Right behaviour. But I will use as much pressure as needed if/when I think it necessary to get a response.
Eg. never allowing her into your space without invitation, whatever you decide that is, whether one foot away or arm's length. Be consistent about consequences & as horses learn by instant association so you need to dish out the punishment/reinforcement *at the time of* the behaviour you want to affect.
If her backup's no good & there's no physical reason, you can bet there are other basics she doesn't 'get' too. I also wouldn't call her 'halter broke' if she doesn't yield backwards to halter pressure. Yielding any part of their body in any way when asked is the foundation of virtually everything we want of a horse IMO. I teach them to yield from fingertip pressure, stick/whip pushing or tapping, to halter/lead pressure, and then to bodylanguage, pointing, directing, waving a stick or rope. I teach them to yield whatever part of their body I direct pressure at - pressure behind them for forward, in front of them, on their nose or chest for backwards, on their neck/shoulder to yield forehand, on their rump or flank for hindquarters, etc, etc.
So, after all that, for teaching backing up, you could start with the horse led into a corner or such, so there aren't many options, 'ask' with light pressure on the lead, &/or fingertip pressure on his nose &/or fingertip pressure on his chest. If he doesn't yield to this, increase the pressure. If/when necessary, you can use the end of a stick/whip or a hoofpick or such as Foxhunter suggested, you can apply sharper pressure that the horse is more motivated to move from, without having to do it too hard or painfully.
It is very important that*The INSTANT* the horse *begins* to yield, release all pressure & reinforce the behaviour. They learn to yield because that behaviour works for them, to relieve the discomfort/pressure. If you have your mind on getting 10 paces of a backup & the horse isn't up to that yet, he'll yield a bit to the pressure, you keep pushing, so he decides yielding doesn't work - he might just try pushing against it instead. The pressure provides the motivation, but the release &/or reward is what teaches & reinforces the lesson. After the horse gets the idea, then you can gradually ask for more.