It sounds like you have already started moving her in the right direction.
There is one thing I kind of wanted to point out, not that it matters in terms of how you change her back . The reason her back is stiff is that she has been 'firing" those back muscles TOO much. They have been contracted , thus the head high and the back hollowed out, for so long that no doubt they have literally shortened. To lengthen them you must make her contract the opposing muscles, which would be her abdominal muscles. That, and such things as carrot stretches which forcefully stretch the shortened back muscles help. But by building up the under side of the horse, the topline develops.
So, the horse must engage those abdominals when transistioning upward, transitinion downward CORRECTLY, going uphill engagee long and low, backing up, backing up a hill, going over cavaletti, walking engaged on a circle with focus on tracking up and under with the inside hind.
Lots of bending laterally will really help. Old fashioned cow pony type disengagements of the hind quarters will help soften and teach the horse to follow the rein, too.
I liek that you see and feel the issue being a stiff back, but focus on strengthening the abs to improve the back.