I see from your profile that you are in Kentucky. Your decision on where to take her will likely depend on a number of factors: her age, your geographical location and willingness/ability to travel, and your insurance coverage. If you want to go to a hospital with an actual dysautonomia clinic (that is, with comprehensive dysautonomia testing capabilities and dedicated autonomic specialists), your options are limited. From Kentucky, you're likely looking at Vanderbilt (Nashville) or one of the two hospitals in Cleveland with dysauto clinics. Your choice between those would depend primarily on your daughter's age, as Vanderbilt does not take patients under age 18. The links to contact information/websites for those hospitals (and the rest of the ones in the US as well) are above in Halbash's post.
If you are limited by insurance, travel considerations, or just want to try a local hospital first, then try to choose the largest teaching/research hospital in your vicinity that is most well-known. From there, seek out the neurology and/or electrophysiologists' (within cardiology) departments. It is often helpful to call and outright ask if they can ask around within the department if any of the doctors have knowledge/experience in diagnosing/treating autonomic/orthostatic intolerance disorders so that they can pair you with the most suitable doctor right off the bat.
Christydrake, above, is one of our "in-house experts" on pediatrics as she's been through the ringer with her own child and finally found great care in Cleveland, so her advice comes from MUCH experience and I give it my seal of approval, for what that's worth. I, personally, get my major testing stuff done at the other dysauto clinic in Cleveland (at the Cleveland Clinic) and then take care of my routine care locally (where I'm lucky enough to have another of the nation's top ranked heart hospitals).
Let us know if there is more information we can get for you, or questions we can answer.
Best, Heiferly.
Dysautonomia and Autonomic Dysfunction are typically diagnosed by either a Cardiologist, or Neurologist. Unfortunately, diagnosis can be quite elusive. Here are a couple of links to our Health Pages that maybe helpful to you:
Diagnosing Dysautonomia:
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http://www.medhelp.org/health_pages/Neurological-Disorders/Diagnosing-Dysautonomia/show/827?cid=196
Dysautonomia Specialists:
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http://www.medhelp.org/health_pages/Neurological-Disorders/Dysautonomia-Specialists/show/717?cid=196
We go to University Hospital in Cleveland and see a pediatric gastroentrologist and neurologist(husband/wife) who specialize in autonomic dysfunction. They have been great!