All righty. Assuming that your position is correct, I would say that you just back up to looking for the smallest give that she's willing to give you. Don't start out asking her to bring her nose all the way to the girth. Pick up a contact, brace your holding hand on her withers, and if she moves her nose toward you even a bit, drop the rope like it burned you. Instant
Release. The release is what's telling her that she did good, so the faster and easier it is for her to feel, the quicker she'll associate the release with the give.
If she does start falling asleep hanging on your hand, I would bump the rope a little to wake her up and remind her that you're still there. If she holds for over a minute with her neck still straight
, bump the rope a couple of times, but don't let slack into the hold. Slack = release, and she hasn't earned it if she's bracing. If her neck is curved
and she's bracing, just not coming any farther, you missed your window to release. Next flex, recognize her best effort and release then instead of holding out for a bigger bend. In the words of Dennis Reis, reward the smallest change and the slightest try. If she's new to the flexion, she's going to be stiff, and hence not going to be "good" at the flexion until she learns the aids and stretches her neck muscles out.
Even what you describe as her good side, 5 minutes to get a bend, is far too long. There should be a pick-up, a following give (in the case of a stiff or confused horse, some resistance and eventually a give), and an instant release of contact. There might be >5 minutes of "discussion" over the flexion, but 5 minutes of everyone standing still and bracing is too long. Personally, I generally fill 5 minutes time with 10+ flexes, 2 ways, taking my time.
Another thing that might wake her up a bit is to disengage her hindquarters while you're asking for the neck flex, but that can get tricky to coordinate.
Glad to hear that you're taking it slow and working on solid basics with her. Good luck!