This article is from 2002! But it touches on this and you can certainly try to contact the two members who posted. There is a support group listed in the post as well. Survivors are the best to talk to. They can really help.
Ok just noticed you are already a member of Encephalitis Global. :) That's good. Let's see if I can't find you anything else that may help...
This one is a bit technical, but it shows some alternates for treatment using a combination of meds to single meds. Although the authors seem to not know whether or not there is such a disease/disorder as Hashimoto's Encephalitis. I think they perceive the variations of the condition to be too vast to be considered the same disorder although the study shows results proving otherwise. But I say, "Welcome to the Wonderful World of the Thyroid". The symptoms are so vast among current patients that most of us are treated like we are psychotic rather than having a treatable condition. So take this one with a grain of salt, but it does have some good medicine info.
--I hope this helps for now. I will keep looking but a lot of the info out there is either the same or there isn't much to report...3 lines or less. So I can see why you are at a loss. I think the more people you find out there to tell you what they went through the better. I would post in the Pediatric Endocrinology Expert Forum and maybe in the Neurological Expert Forum and Community. There might be a few other Forums on MedHelp that would be of some assistance to you and it's worth posting your thread multiple times to see what you get in response. Maybe another member in this Forum can help too. I really hope you get the answers you need soon. Your little girl deserves it.
You all are in my prayers.