Cockatiel pulled stitches out of his eye lid

My cockatiel cut her lower eyelid and recently had surgery to stitch it up. This was about a week ago. However I just noticed that tonight she has pulled it and actually undone the stitches so her eyelid is hanging lose again. She cut it about a third of the way in from the outside corner.

I'm very worried because they had to cut off a section of skin to stitch it back up and I'm not sure if they will be able to do it again.

When I took her in the vet said she has to have an operation and they 'will try to save it' but I don't know if they meant her eye lid specifically, or meant she wouldn't be able to live without an eye lid.

My question is basically, if they can't stitch it up again can my cockatiel live a happy life with her eyelid split? It doesn't seem to bother her at all but I don't know if it will hinder her at all. I'm taking her back to the vet tomorrow but I guess I want to have as much information as I can get. Thanks!