I know shaving is a real turn off for women and girls, but James is telling you true. All these other temporary methods leave you flirting with bigger problems in the future. One word of caution though: Do not do laser on her face. The hairs are too fine. It hurts a lot. It may induce more hair to grow.
If I were you and if she is highly motivated, I would think electrolysis. It must be gentle and the practitioner must be motherly (or fatherly) and reassuring. I work on younger teens, when THEY ARE DRIVING THIS EFFORT WITH HEALTHY MOTIVATION. Ease her into this. If you can find someone with a newer computerized epilator, sensation and speed will be on her side. Someone with an older epilator who is experienced can come close sometimes to mimic what a computerized epilator can do.
Just get those consults. If you can personally go check out people and have electrolysis performed on yourself first, then you can spare her from being the guinea pig as you decide on who you will trust with your beautiful daughters skin. If she see’s that you are having electrolysis and you are positive about this, it may be easier for her. Daughters watch their Mothers closely as I’m sure you know by now! Prepare the way for her. If you have to travel a bit, turn it into something fun as well. Do something in combination with the electrolysis treatment, like have a lunch together in a different a kind of restaurant. Go shopping. Go sightseeing. Combine it with a movie. Start the search close to home and spread out to a travel radius that is doable for you.
Good luck and work hard to make a proper decision.
Dee