I would reccomend any book by Martin Seligman, if they haven't already read him, such as 'Raising an Optomistic Child.'
A style of therapy called 'Cognitive Behavioral Therapy' may be useful.
PG kids can have a very hard time, and it's heartbreaking.
Seems to me that a service project that the DD can really sink her teeth into would raise her belief in her self, if she can manage something that interests her.
I'm wondering if the school could manage to have a place for her to get 'little rests' in, as needed, perhaps a nurse or special ed teacher, so that she can attend even when she is sick.
Weirdly, I'll bet it's not to late to start 'praising effort' instead of inborn traits, particularly when the child can overhear 2 adults discussing her, rather than to her face. I'd even suggest that you advise the Mom to start praising her own efforts aloud, as a model.
Lately we've been saying things like:
90% of life is just show up. Now it's time to show up.
We know that it's been very hard, and that things haven't always gone well in the past, but we believe that if you keep trying, sometime soon, you will find that the strength you need is inside.
over and over and over and over again.
Some people, especially gifties, seem to 'not get' that life is often really hard. A friend of mine says that he enjoys Buddism because it taught him that 'life is supposed to be painful.' (not sure if this is true or not, but it's interesting.)
Lastly, if she has recent enough test scores, I would highly reccomend that the family apply for Davidson's Young Scholar Program. Spending time with other families in similar situations would be a great relief, I'm sure. At the very least, see if your friend would be willing to sit down with you and read some of the posts here. A surf party!
love and more love,