If your abs are hurting you are probably a big air junky doing board grabs which basically means lifting your legs with the weight of the board. sit ups will not get enough ab strength for that as you need to lift beyond body weight. do leg lifts & 20 variations & lots of weighted ab exercises. also the jumping move is rapid ('explosive' in gym terms) so forget slow steady and think of a power move. think about what good session would be: 100-200 jumps? So can you do a similar set of weighted explosive ab moves in the gym? Also, cardio is probably not going to help. yes your heart rate is up kiting but I've never felt that to be the limiting factor.
Your abs will hurt because kiting is a sport that uses your core muscles and not specifically because you are doing lots of jumps. Kiting is basically one big sit up and improving core strength will help with endurance aswell as improving your stance which will also help reduce fatigue.
Avoid weighted weighted abs exercises unless you are quite advanced as you can do yourself an injury and as your core muscles are actually a lot of small muscles in groups it's technique rather than weight which will give the best results. Body weight is all you really need when doing core training with very light resistance if required to make the exercises more focused ie medicine balls, bands.
Plyometric training (explosive action) is extremely good for fitness but more suited to sports where rapid increases in movement are required, probably better for wake style style of kiting.
Although kiting isn't as reliant on good cardio fitness like running or cycling it is important as it is your overall level of fitness and if you are unfit then you'll be struggling before you even hit the water and your body will take longer to recover from the days kiting.