My child fell on her lip, which is swollen and open, not bleeding but very sore?

If the cut crosses the lip border she needs to get it sewn up so scar will not affect her great smile.

If it's gaping open then it might need a stitch or two to close the wound. You can just apply cold compresses to area to reduce swelling and let it heal by itself. This way often gives a better cosmetic result than stitching does, but it takes a little bit longer to heal together than if you stitch it. I would opt for reducing the swelling let mo th her nature heal the lip.

Swollen lip: You can give ibuprofen or Tylenol for swelling and pain. The mouth is rich in blood supply , so it should heal quickly on its own. To the emergency room if the wound is gaping open that may need stitches or it does not stop bleeding.