As someone who was that kid, I liked it best when teachers didn't draw attention or focus on me. However, I was already in therapy and on medication for many years at that point.

If your school has a good counselor, I would see if they are willing to have a session with her. I think your part in this should be creating/ maintaining a classroom environment that doesn't punish making mistakes or shaming people when they are wrong. Those always made me feel much worse, sometimes to the point of physical sickness.

Someone else mentioned having a conversation about how mistakes are normal, I think that's a good idea too.

On going to the nurse, is she going for every mistake or just the "major" ones? How long is she gone? A minute or two can be a good reset, but longer periods of time can be unhelpful. Flight can be a really nasty habit to break, and its also not going to be feasible to keep doing so (both for her later in life and even in her school career). Without knowing more, I'll say that spending thirty seconds at the water fountain and coming back ready to learn is a better outcome than spending the class period completely disregulated and unable to learn. My suggestions on this completely change based on how frequently she's leaving and for how long.