Everyone else already gave you the basics, just adding a couple thoughts.
It is your ISO. If you are happy with grainy pics, it's no problem. I took the below photo at a friend's show, and yes the entire shoot is horribly noisy, but I'm okay with that especially at smaller sizes:
Note that redicing the image size will help smooth out the noise, so you could resize and then try a light unsharp mask. There are noise reduction tools, too.
Other than that, you should get off of auto ISO first. If she can stand still for you, you can drop the ISO and the shutter speed also to compensate. Going to ISO800 would only require you to double the shutter speed to 1/20, probably needing a tripod but not so slow that she can't stay still long enough for a clean shot. ISO400 would mean 1/10th, and you might start running into problems there.
A faster lens would surely work, f/4 (one stop) would let you drop to ISO800 as well, and f/2.8 (two stops) to ISO400. Below f/2.8 I think you'll run into issues with too shallow a DOF, if her eye is in focus something like her ear might not be.
Good luck, I think you may, at this point, just have to learn to love noise.