D.B.
No you don't go to the ER with a sprained ankle. It's awfully hard to see our kids in pain, but really, she ran around on it so it's not broken. You did the right thing with pain relief and peas, and I think probably it was made worse with the hot bath and the running around in the park. The key to sprains and similar injuries is RICE - Rest, Ice, Compression, Elevation. This is what the ER would tell you. So an ace bandage or ankle brace would be good (from the regular pharmacy - ask the pharmacist if you need help), as would keeping her foot elevated to reduce the swelling. Pain relievers are fine if there is no question of bleeding.
I'd see how she is in the morning, and make a decision about school then. She may need to stay off it for a day. If it's really not doing well or there's excessive swelling, call the pediatrician and ask if she should be seen for an X-ray. But again, this is not an ER situation. Hard as it is to believe, even a broken bone isn't a dire emergency other than the discomfort angle. And I'm guessing nothing is broken if she ran around for two hours.
RICE.
At some point, if she needs to be seen, I'd take care of that and then report it back to the school just because the injury took place on school property. They'll record it, and probably their insurance policy will cover your out of pocket expenses (copay, etc.).
Sorry to say that childhood is full of injuries, and you just need some experience in learning to figure out what's worthy of a visit and what's a "deal with it at home" problem.