Hi, so sorry you have had to come here with such a horrible experience.
I am no doc, but from my own experience I would say that the symptoms you are describing sound very much like CRPS. However, the docs are trying to exclude other explanations, and that does take time - time you just don't have when your child is in so much pain.
She needs to see a pain management specialist as soon as possible, so that she can be diagnosed (whatever it is) and treated quickly - the early months are crucial. In the meantime ask her primary care doc to consider starting her on some form of nerve pain med like Gabapentin, amitriptyline or Lyrica. If it helps then that's diagnostic for them as well. You could take in the Wiki page on CRPS for them to read, and ask their opinion. Ask too for something like a pain relieving patch (lidocaine etc) which might help. CRPS pain is no joke, and many ordinary painkillers just don't work - things like tramadol etc might. You might need to push hard to get something unfortunately
Try to get some unperfumed Epsom salts (chemists, health shops, online) and use a few tablespoons in a warm bath. A lot of us use them to help with the pain, they work by blocking calcium channel receptors (something like that, sorry if science slightly ropey!). If she can't tolerate a bath then try wrapping some in a damp cloth and placing it gently on any burning areas.
IF it is CRPS then it's important to keep things moving - CRPS thrives on non-use, so even though it hurts she needs to gently move as much as much as she can bear. Heat packs might help, but make sure she has something soft between her skin and the pack, and keep an eye on the skin, don't leave it on too long.
Vitamin C 500mg twice a day is a good idea, it has been shown to reduce the risk of spread and many of us take it preventatively.
Relaxation is a big thing too - her body will be completely braced against the pain, and the stress is very bad for you. It's hard, but try soft lighting, calm music, nice smells, and meditation too (tracks on iTunes) - I went from skeptic to believer myself

but it really helps with the pain. I didn't realise how tense my body was until I consciously relaxed. You might find it helpful too - it must be awful to watch your child go through this and not be able to help.
Please tell her that there is hope, and help out there, and wish her well from me. I send her a big gentle cyber hug

and please do let us know how she gets on. She's lucky to have such a supportive mum.
Bram.