I'm sorry to hear this. I cannot for the life of me understand why the text fields in which her Apple ID appear will not respond to her touch unless the shaking of her hands has some nearing on this. But what wouldn't make sense about that is that she was able to tap the ID and sign out.
I Don't want to make the situation any worse, but I suppose that she is sort of at a dead point now anyway, if she cannot sign in. Have you had her try to reset all settings. Settings>General>Reset>Reset all settings. She will not lose any data. The iPad will reboot on its own and she will have to enter some of the device settings again. IIRC, she will have to enter her WiFi network password again so if that's a problem, forget about resetting all settings.
I Just noticed that you aid this happens to you on occasion as well on the Mac. It must have some connection to the .me suffix. Something else just struck me and that is I was under the impression that you could not change an Apple ID with a .me suffix since those are already Apple ID's. I never had a .me email address.
DId you read the first part of this in article? I assume that you have read it, but since the .me is already an Apple ID, could that be why it wants to keep reverting to that in the settings?
Change your Apple ID - Apple Support