I really feel for you and your DD but please know that both of you can hold your head up high
The 11+ system for a superselective in particular, can be very brutal on parents, let alone a little one of 10, or possibly 11. You both lived through an incredibly stressful and worrying time and have come out the other side a little battered, to say the least. However, having also had a DD who worked hard and didn't achieve the result she wanted, I do know that in time, you can both feel so much pride for having the strength to approach the 11+ in the first place.
My DD1 knew she wouldn't "pass" but she wanted to go for it, none-the-less. Knowing this though, didn't stop her feeling "gutted" when the result came out on March 1st. The thing that we clung to, was that no-one could take away the boost her education had been given, by preparing for the 11+ exams back in November the previous year. Her confidence in her abilities had grown immensely and she flew from then on. I realise that the results now coming out in October makes my anecdote a little lame but I wanted to say that nothing is wasted. My DD went on to achieve everything that she wanted to...and still hasn't reached her optimum potential. The 11+ was a turning point for her - she suddenly started to really focus and believed that if she could do that (after years of SENCo help) - she could do anything.
I truly hope that in a few weeks, your daughter will get her confidence back, she deserves it. Could you perhaps focus-in a bit more on her out-of-school activities for a while, to boost her confidence? We did that with DD1 - she did a competitive sport and so we spent more time doing that, so she could feel more comfortable and build her self-esteem back up.
Your daughter is amazing for even having sat her bottom on the seat on 22nd September 2012. She's brave and
bright! She achieved a score of 308 which certainly shows she's intelligent!
I'd pretty much agree with what moved has said, in terms of what you tell her and I agree with what moved said about the 11+ and GCSE results due to my personal experience with my DD1. As an ex GS pupil and now a parent of two DD's who have sat the Essex test, I have mixed feelings about the 11+ exam system. Even those who get what they need, don't necessarily come out of it unscathed or untouched by the pressures of the entire process.
I wish you and your daughter peace at this very painful period in your lives. My heartfelt thoughts are with you both.