The kip is a difficult swing dance aerial to get. Because you can only practice it maybe two or three dozen times before you have to call it a day, it tends to make people shy away from it and focus on something easier. I love it though, and I'd love to see the entire swing community be able to get it. That being said, I have some suggestions on how to train for it in the beginning. I'll break this into three subsections, main types of exercises that build primary muscle groups needed to execute the aerial, stabilizer exercises that build strength so you can adapt to the changing dynamics of the learning process, and suggestions on how to do the exercises to gain the desired result.
Main exercises :
Leg Presses or squats, bench, military press, push ups and inverted push ups.
Anything you can do to build strength in your torso.
For your stabilization exercises, just do them as normal to build strength. Tone doesn't really matter, strength does. You have to think about what's going to happen when you're needing to redirect that momentum. If she hits you at 10 degrees off, it's all your stabilizer muscles that redirect her onto the right track before explosive power can kick in. In the beginning, you'll hurt like you just got in a car wreck because you're working every stabilizer muscle in your body. I assure you that after you and your partner get some technique training under your belt, you'll need to use these less and less, and your power will come out.
For the main exercises, lift for explosive power, not for strength. What does that mean? Say you're on the leg press machine. Rather than doing slow reps of a lot of weight, stack as much weight as you can stand on there and try to push machine into the next room with each rep. Think "on and of" and at full speed. Follow through, and literally push it so fast and with so much of explosion that it leaves your feet. When I get on the squat machine, I literally try to put as much weight as I can onto my shoulders and attempt jump a foot in the air. That is exactly what the kip feels like when you're working with a new partner. Same goes for bench and military press.
As a general rule of thumb, if you can bench yourself and your partner's body weight, you'll get it quickly so beef up. If you bench 150% of your weight, you're going to have to rely a lot more on technique. If you can only bench your weight, it's all technique and it's going to take a good amount of time to get. So, strength training here really makes a difference. The more strength you have, the less you're going to need her to jump. And in the beginning, most girls have no idea how to use their legs to provide the upward thrust necessary to kip.
Main exercises :
Jumping high, the same exercises I listed out for the men above, and if you have the ability to, drop some weight in a healthy way (*** see explanation ***).
Crunches, russian body builders, flutter kicks, and anything you can possibly do to strengthen your abs.
In the learning phase, I think the single biggest thing you can do is working on jumping high. Leg presses, running, jumping rope, whatever you can do to build up your legs. If you can dunk a basketball, throw a triple lutz, or do a punch back, you'll get it pretty quickly.
Why? For just half a second, at the most critical time of the kip, all your weight is on one foot and you have to jump straight up in the air. If you can push your center (hips) all the way up on top of your partner's shoulders then gravity will take over and the kip will work as soon as you figure out how to aim your jump. If you have this much vertical leap off of one foot, you can kip yourself.
Think about what that means though, to get your hips up to your partner's shoulders. When training, forget about what your feet are doing, just think about your hips. You know you've got some jump in you if you can stand straight in front of a wall, mark where your belt buckle is, jump, and get your belt buckle two or three feet higher than your initial mark. That's the kind of lift that'll make it easier for you to do this aerial.
After that, there's all sorts of techniques you can do with your platforms that'll help you control it, so don't forget about working your shoulders and abs.
*** The comment about dropping weight in the beginning should be explained. After you have experience with many of the techniques of the kip, weight is almost negligible compared your ability to jump and use technique. In the beginning though, your weight will be a liability no matter how much or little you weigh.
There are two reasons for this comment and the first relates to how much you can jump. Think about your leader, if you only have one foot of vertical leap like more than 90% of the people that take my kip classes, well, just know you'd really need about three times that. Your leader is basically going to be doing about 2/3'rds of the kip using his own shoulders and legs. It's simple physics. The lighter you are, the easier it's going to be for him to throw you.
The other reason for that comment is geared toward female athletes, maybe lifelong runners, yoga teachers, etc. It has to do with the misuse of power. Say you're a 120 pound solid muscle lady that can jump four feet in the air. You may be a dream aerialist of tomorrow, but you're going to hurt more than you know on day one. The first time you really hit a strong power jump and go off axis or off connection even just a little bit, well, let's just say even the toughest of men can't handle the kind of pain this aerial can cause if it's done wrong. It's sort of like swinging a rocket on a rope. Your kinetic energy goes in that one direction until that direction is changed, and that of course means the leader is going to have to either absorb that energy or compensate for it with stabilizer muscles. Again, it's all simple physics. Your weight used in the wrong directions can easily cause a leader to have to ice his arm or shoulder.
Good luck all.