Oh, wow! It was really moving and inspiring to hear that she pulled herself together to face another struggle. Your daughter is clearly a great kid, and sometimes this is just too much for her.
I'll give my experience with regard to needle pain, in case it helps:
Through experimentation I've found a lot of areas on me that usually don't hurt, and some that are likely to REALLY hurt.
The MAIN thing is, when I'm really stressed, when I've just had an argument with my girlfriend, or had a bad hypo recently, etc., almost anywhere hurts, and it hurts a lot! I will try 4 or 5 sites until I find one that doesn't make me jump or cry. A suggestion: sit her down have her take 20 deep breaths and let her arms and legs relax before trying again. Also, if you find an area that is usually painless, save that area for the most stressful moments.
I always stand up to inject, it seems to hurt more sitting down. I guess standing up stretches the skin more.
If she hasn't yet learned to inject herself, maybe she would prefer this, as it's easier to avoid pain if you're doing it yourself.
All the best!
EDIT: Oh God, I noticed hypo symptoms just after writing this, and I was at 2.6 mmol/L! Treated immediately but now I feel vertigo and anxiety and like my veins are full of metallic sludge. Curse this disease.