I echo all the other wonderful comments here and just wanted to add my perspective from your daughter's viewpoint. I've had stitches 4 times before the age of 12. I hope your daughter won't need to get stitches again, but sometimes parents just have to accept you have clumsy/feisty/adventurous kids.
Please do not beat yourself up. Regardless of who is there, these injuries can happen. My mom saw me roll off the couch, which is when I had my first set of stitches on my forehead around age 2. A summer camp counselor was there when I decided to jump backwards into a pool and got stitches on my chin around age 8. A friend's mom was there when I sliced my finger on a swing set in her back yard around 9. And my uncle was watching me when I crawled onto a glass fragment while playing hide-and-seek and sliced my knee open at 6.
I turned out ok and my mom is forever warning my kids, "Be careful!" because she fears they'll be like me. No stitches for either kid yet, but we've had a broken ankle, knee sprain, ankle sprains, and other various ailments. Life happens!