Great going on the house.
Sell the pony. Bad attitude is nothing you need with all you are doing. And while you might be able to retrain it, there is no guarantee it will behave for the kids - just for you. Hiring a trainer may be an option, but why spend the money if you need it for the house? Same problem with having the trainer work with it. Unless the rider is experienced there is no guarantee that the pony will be a good ride for kids. Sell this pony and get another child safe pony for the kids.
It already threw you and you could have been badly injured. Even a broken arm or leg is bad with as much as you have to deal with, next time it could be worse. Bad attitude is nothing to fool around with in a kid's pony or horse.
Unless this is a fabulous show horse worse spending a lot of training $$$ on, don't waste the money on the trainer, cut your losses and get a gently pony for the kids. There are a lot of good horses and ponies out there already trained.