So you much surch everything (that causes anhidrosis), that includes hypothyroidism (test for TSH, free T3 and free T4), autoimmune disease (sjogren's), diabetes, metabolic disorder, skin damage, genetic factors, nerve damage, etc.......so first I would go to a Ped Dermatologist to see if it's a problem of the eccrine glands (no sweat glands, sweat glands malfunction, etc) and possible tests you should do for the right diagnosis (of Anhidrosis) . The doctor will ask for many tests, and you can also ask for the thyroid analyses, etc There is also a Ped endocrinology expert forum here, you can post and see what is the opinion of Dr. Guy.
I don't know what the cause is. Not sure where to start. I read your link...not her. She only had maybe one symptom. I actually see a little sweat form on the bridge of her nose..but tha'ts it. Her skin is just so extremely dry. I just don't know where to begin to help her.
What is the cause? Treatment of anhidrosis involves addressing the underlying cause (thyroid, etc)
Probably she have a inherited disorder (genetic). Found this (if this is the cause, it can help):
http://www.socialstyrelsen.se/Sos/ExternalWeb/Templates/CommonFulltext.aspx?NRMODE=Published&NRNODEGUID=%7b12956576-198F-4220-976F-E6BD69235D8B%7d&NRORIGINALURL=%2fen%2frarediseases%2fHypohidrotic%2bectodermal%2bdysplasia%2ehtm&NRCACHEHINT=Guest#7