You are not letting her down. We all want the best for our dc and it can feel overwhelming when you don't know how to fix something.
We have had this problem with ds(14) at various stages, the "I can't do it" when you know they can and are either lazy or not thinking it through. Tears or sulking when things don't go their way or they lose a game.
I am not the type to give up either, but just couldn't pass that onto ds and dh was a big softey and always did everything for him. When he was younger my dad was nasty grandpa, who he loved, and would cheat/win games and gave him determination for a while, but after he died it faded away.
Karate helped for a while, along with a competitive sport he was ok at. But when he didn't give his all and his team mates/friends developed quicker than he did, he was dropped from the squad and he sunk back to not believing he was just as capable and could do it if he really wanted to.
He said he wanted to do it and I tried to work out a home fitness program with him but 5 mins on treadmill or 10 sec planks or 5 squats and it "was too sore", "can't do it" and sulking. We splashed out quite a bit and got him a personal trainer for 10 weeks just to teaching him how to do squats etc properly and the difference from day 1 was amazing. He does 1 hour a week with this guy and it influences the rest of his weeks attitude towards everything, fitness, diet, home exercises and even school. We are going to do another 10 weeks and then see if we can drop down a bit as it does cost a lot.
At 9 a PT is way OTT but it shows that sometimes they just don't want to listen to their parents and you need outside help to boost their confidence to try more. Does she have a relative that can take her somewhere and do things, my SIL was always amazed at what my niece could do with me but wouldnt be able to do at home. Have you thought about an activity that could help? Martial arts, drama etc?