To get out of the hair in you baby's eyes safely, you'd better clear your hands totally so as to prevent bacteria from your hands from getting into her eyes. Then, you may need someone to help you hold your baby( don't allow her move her hands or legs to effect your "work"). Then, can pull eyelid gently and get out of the hair. Hope it will works.
Well, getting a hair out of the baby's eye can be kind of dangerous. Because your baby don't want you touch her eyes, it can hurt her eyes when you try to take the hair out. Also, baby's eyes are so sensitive that you may get them infected if you don't wash your hands carefully. So in my opinion, you may not be supposed to get the hair out just by yourself. You'd better call someone else to help you or just go and see a doctor. I think they have some advanced machines which can get it out in an easy and safe way.
You'd better find another one to help you hold the two hands of your baby to avoid the baby's movement. Then, you must wash your hands in order to make sure of its cleanness. Next, you'd better cut your nails to short in order not to scratch the baby. Last of all, you can do gently to get the hair out of your baby's eyes.