Ah, The Dreaded Axel. (cue ominous music).
There seem to be more tricks and angles to teaching axels than anything except the mohawk. Like the mohawk, the complexity of the skill takes it to a new level. Both of these are skills that break with the basics before them; they are watershed skills. But where the mohawk is a basic skill that emulates natural positions, there's nothing terribly natural about axels. You are making your body do something new with this jump.
The biggest impediment to quickly mastering this skill, however, is not its physical aspects. It's the mental aspect. Even really talented skaters can take months to master this skill; some of the best skaters I know took more than a year.
And once you've been working on this jump for one or two...or three...or four...OR MORE SESSIONS AAAARRGGGHHH, it takes on mythic qualities. It consumes the skater and her mother, and they start to assume that everyone is watching, that everyone is talking about how Suzie is NEVER going to get that axel, she might as well switch to ice dance.
Or move to a dessert island in shame.
Truthfully, no one is talking about you, and in fact if you keep your mouth shut, no one has noticed that you've been in Freestyle 5 for 3 sessions. It's important to remember that the axel is not the only thing in skating, and that literally everyone who keeps working on it does in fact eventually land it. This happens on the principal that if someone can do it, it's possible, and if it's possible, you can do it.
It's also vital that the axel not be the goal. You can do plenty in skating without an axel. You can compete "up" in USFS nonqualifying competition-- you won't have that Preliminary test, but you can compete at Prelim without it. ISI now has the Open Freeskate levels and mixed level Artistic events so that you can use your solid Freestyle 5 elements in competition without the axel. And yes, there's ice dance, and synchro, and Moves tests (which you can move through independently of freeskating tests).
Here's what you cannot do: you cannot insist that an entire program or ice show be changed in order to accommodate your skater, who only wants to skate once a week, because she can't get the axel. You can't complain about the "gift" or "pity" passes that other skaters got, when your skater's axel is better than theirs (it's not, trust me). No one's giving anybody pity passes.
You can't switch around from coach to coach looking for the one with the magic wand. Again, trust me, they are all telling your skater the same thing.
And mostly, you cannot invest this jump with all-consuming importance. Believe me, I've been there. It was during the axel-hunt (we're talking 2 1/2 years, from a skater who ended up as a professional show skater) that my daughter banned me from the rink-- wouldn't skate if I was watching. She took ownership, got rid of my noise and anxiety, and finally nailed that jump. Thank god for the wisdom of 10-year-olds.
You will get through this. Your skater will land that axel. Stop worrying about it. I mean that literally.