As long as you have ruled out pain issues, it's just a matter of time and repetition to get her comfortable moving in the frame you want her to carry herself in. Transitions from walk to trot and back to walk will help her get the idea to hold that soft, round frame she does at the walk, while trotting.
Start by lunging her along at the walk with her head down and back rounded up for a dozen strides or so, then ask for a trot. Yes, her head will pop right up and her back will hollow out, so only trot for a couple of strides and then immediately
take her back to the walk. Round her up again at the walk and let her calm back down. Once she's calm and rounded up nice again, ask for a few more strides of trotting, then immediately
go back to a walk again. Pretty soon she will figure out that she is only going to be right back to the walk and rounded up again, so it's much easier to just hold the rounded up frame, since she's going to have to walk and round up again anyway.
Quit the session on a good note and go do something else. Come back to it again later. Work at it each time you have her out. How quickly you can progress will depend on the individual horse and how quick they figure it out.
Oh yeah, don't make a big deal out of "punishing" her, just bring her back to the walk and round her up every time she gets her head up or hollows out her back. Then ask again. Just keep building on each success.
Once she has it down at the trot, then do the same thing with the canter/lope.
As for how long you should work her, just keep an eye on her. Don't keep going until she is breathing really hard or is sweating to the point of dipping foam, but good rapid breathing and a nice even sweat is just fine. Give her breaks to just stand still and catch her breath. And always be sure to let her recover to normal breathing and a cool body temperature before putting her up.
Another great way to get her using her head, neck and back is to work her over things that she has to be careful with and watch her own feet. Go for a walk around where you keep her (just by yourself, don't take her along the first time) and find some safe obstacles you can work her over and through. Find places to take her out and just see the sights, walk her over downed branches, send her up and down any little hills you may find, up and down through small ditches (check for safety first), if you have a small creek or stream, send her back and forth across it or walk through it while you walk alongside. Play with each obstacle for a bit and move along to something else. It all helps. And it's a lot of fun too!