I am Italian/Puerto Rican (dark & hairy), and when I was 12 I used a bleaching product. I have to say, I don't recommend it for a 9 yr old. I was never really happy with the results anyways, because instead of a dark upper lip, you can get an oddly light patch above your lip from the bleach.
The ingredients in these products are not safe for a child (or anyone). Your skin absorbs up to 60% of what you put on it, and it gets absorbed directly into your bloodstream. At least when you ingest toxins in foods or drugs, they are filtered by your kidneys first. Cosmetics go right through your skin and into your bloodstream, not to mention the horrible fumes of bleach you inhale when it's sitting right under your nose for 5-10 minutes. Consider that people are now using birth control patches and nicotine patches- even pain killer patches in hospitals- as science has learned how much we absorb into our bodies through our pores. Back when all these products were developed, people thought skin was more like shoe leather and didn't really take in what was applied to its surface. The FDA doesn't regulate cosmetics with the attention food and drugs get- so YOU must discriminate and educate yourself on what is safe. The basic guideline would be- IF YOU WOULDN'T PUT IT IN YOUR MOUTH, DON'T PUT IT ON YOUR SKIN!
I just pluck any unwanted hair- safer, cheaper, effective. Plucking can be painful at first, but if your daughter is bothered by the hair, she will not be deterred.