I'm a therapeutic riding instructor and have more than a few nervous students. I've found making games out of it takes their mind off the fear and onto the challenge of the game.
It's aright for them to be afraid, telling them "don't be afraid" or "that's silly to be afraid of" is putting down their fears and denying their feelings. They are honestly afraid, so be careful to avoid saying things like that - it's hard to tell when you're doing it, so keep that thought in the back of your mind.
Some games I use to encourage my students-
Obstacle courses, My favorite starts with a pool ring on a barrel, they pick it up (put it around their wrist), weave through a set of cones, then over 3 poles, then down the 'horsey freeway' where the 'speed limit = TROT!" then at the end of the long side is another pole a "speed bump" to remember to slow down to put the ring back on the barrel. It gives them a targetted job to do, keeping them focused on the task, not completely focusing on the difficulty or speed of riding.
Another fun thing to do, if the horse is bombproof is "jousting" the student holds a skinny pool noodle, and you (on the ground) hold up rings, all over the arena, the student has to walk or trot by and catch the ring on the pool noodle. Be sure to help them along if they have trouble to keep them encouraged.
You could try having her practice fake barrel racing at a trot to increase her confidence steering while trotting, if she's comfortable, she could canter back to the starting line.
Also ask the girl what HER riding goals are. Many students goals are very different from what their trainers goals are. Ask her what SHE wants out of riding, if it's just fun, make games! If she has a specific goal just get creative, make games to make it worth it for her to do the skill.
As for her being afraid to hit the horse with a crop, I teach my students "ask, tell, command" ask the horse with a gentle squeeze and a cluck, if no response a big kick and a little growl, still nothing, a shwack with the crop and a big kick. Many students are afraid of the crop because they're afraid the horse will lunge forward, not because they're afraid of hitting them. So be sure to reassure her that the horse won't do that - or better yet teach her how to prepare herself for it by keeping her heels down and sitting light to be able to move with the horse.
If her goals are actually jumping, try integrating jumps into the course, do some real easy ones and some medium tough ones - don't make them 'jumps' or 'oxers' or 'verticals' make them a "speed bump" or a "bunny hop" or a "river" or a "fallen log" or "construction road". Most students, if they aren't told how big a jump is - don't really understand what jumps are more difficult, they wont be so scared.
The point is to make it part of a job, maybe she has to help a beany-baby dog make it back to it's family on the other side of the ring. Get creative :)
I also find giving them options helps a great deal - if she's nervous with both cantering and jumping, ask her "want to canter or jump first?" "want to spend more time on cantering or jumping?" and so on. Often having a sense of control helps them approach their fears a little braver.
Sounds like you're going to have fun! These students can be the most rewarding IMO because you can watch them flourish and overcome so much. And remember, make a HUGE deal out of every little achievement, it's a huge deal to her to conquer that fear, it should be a huge deal to you too. Stay positive all the time, getting negative will put her in a negative mood.