I'm so sorry to hear about your daughter, more and more young people seem to be getting this disease, it's crazy! It seems the change in hormones with puberty is a factor - got me it was pregnancy and for others it's menopause.
I'm so glad you want to get to the root cause and figured this out early, for me it took 3 years, it was only when my hair started falling out that I stared to take my own action.
The GPs and Endos will help you with such things as bloods and ultrasounds but that's where it stops I'm afraid, they deal with the symptoms - give you the hormone replacement - but not with the underlying issue. What you need is a Functional Doctor if that is possible where you live? Please read the link I posted on my page - it's my 3rd post I believe. I have started working with a Functional Doctor here in NL and at my first consultation we figured out my triggers and underlying gut issues - the gut issues are the most important as this is where all disease stems from. For me it was multiple food intolerances - gluten/dairy/sugar (please note you will have to change your daughters diet), IBS, Candida and EBV - has your daughter ever had glandular fever?
Obviously we are all unique and different factors will be affecting your daughters health. The one main thing that will be there is 'leaky gut' all autoimmune diseases have this in common and there will be an imbalance of gut flora so a diet protocol like GAPs or AIP (Autoimmune Paleo) will fix that. You will have to heal her gut, if not over time other autoimmune conditions will start popping up. Start with baby steps, maybe one week cut out the gluten, see how she feels - Paleo is a great diet (not a diet) to look into for receipes.
Hashimotos is like an onion, you pull back one layer to discover the next problem. Vitamin deficiencies will also be a problem - most likely Vit D, selenium (Brazil nuts are great for this), B12, Iron - again everyone is different but I know there are a few specific ones. Maybe Oestrogen dominance is an issue with the onset of puberty?
Your daughter has inflammation in her body and that is why you're seeing high antibodies, keep an eye on them and hopefully once your daughter starts addressing each issue they will come down.
You are going to have to educate yourself as much as you can, some of the girls above have already mentioned some great Funtional Doctors who have blogs, podcasts, Facebook pages etc and if you like over time I can send you more information that I found really helpful, I don't want to overload you all at once.
Oh Coconut oil - is great for the thyroid, maybe 3 tps a day, it will also help balance your daughters blood sugars if she is having Adrenal Fatigue - another common factor that goes with Hashimotos. She'll probably need probiotics also.
I'll leave it there for now but I'm here to help if ever you need anything.
Sending you my best wishes 😊