If you can't afford a trainer then that's fine, but just be extra careful with your kids when they are riding as you wouldn't want them to get hurt.
I'm not sure of your experience level, but there is always time to learn. Have you called the chiropracter to rule out any back pain issues? Do this as soon as possible. As you said when you applyed pressure to the back she moved in circles so I assuming it may be a pinched nerve, or a back issue; i think you would be please to know this is an easy fix.
If you want your children to gain experience in horses, then let them groom the pony, take it for leading walks, clean out it's feet. Insted of letting them loose confidence on a pony that throws them off(: