My second child was also a "do it myself" person. For 4 years, if she was balking at doing something like cleaning her room, all I had to say was, "Well, I guess I'll do it." She would immediately scream, "NOOO!!! DO IT MYSELF!" (At about 6 years old, she started to realize that maybe there were ways of being stubborn that didn't include doing what she was asked. ;D)
I've always been a "do it myself" person, so in a way I related. But, watching my kids, (I have four, ages 21 to 9) I started to see that a great deal of their stubbornness about things came from fear of being imperfect. That form of perfectionism is especially common in intelligent children - mine are intelligent, and your daughter is, too. They start to feel that because they are so much smarter than everyone else, they should never make mistakes or need help. When they do, it makes them feel worthless.
With two of my kids, we have to frequently discuss how good enough is, often, good enough. My thirteen year old still refuses to do many things unless he thinks he'll be the best one, and he'll be the best the first time he tries it. I've had to tell him, "Even if you're last, that's OK." He loves Boy Scouts, but often balks at finishing a particular reqirement that makes him feel awkward. Right now, he will advance almost 3 ranks if he completes one task, the swim test (jump into water over your head, swim a certain distance with one stroke and back with another.) I keep trying to tell him that he just needs to finish it, even if he thinks he's done badly, and be DONE with it, but he doesn't like the idea of just getting through. He doesn't want to do it until it will be up to his own (nearly impossible) standards.
On the other hand, with my other two kids, I have to point out why slap-dash, hurry through it work isn't good enough. With them, I have to insist on something that resembles actual effort, even if it's not their best work. Everybody's different.
So, the long answer is, she'll always be a "do it myself" person, but you can help her see that everyone needs help (and makes mistakes.) If you have to hire a repairman, an accountant, whatever, tell her, "I really need this help. I couldn't do that job on my own." Point out other adults, especially adults she admires, asking for and getting help. In time, she'll come to see that it's OK to occasionally be less than perfect.