Please sit your daughter down and tell her that both of you can look on this, in the long run, as a positive thing - because it is a wake-up call that her hobby-for-cash needs a serious reboot as a legal, licensed and taxpaying business. She's good at nail art! Now she needs to:
Pay this mom back immediately and completely. Now. No discussions, no holding back any money to cover your daughter's supplies or anything. Your daughter will feel that this is unfair and basically rewards this woman for bad behavior, but if she's going to be a businesswoman, your daughter must put aside the teenager voice that cries "Unfair!" and adopt an adult perspective: Pay this woman to get her off your back.
Stop all nail work now, and do intensive research about what daughter needs to do to get trained and licensed in your state to do nails. This is a licensed business that involves health and safety regulations, and people can be prosecuted by the courts for operating nail salons without a license (it has happened in our state). Your daughter, thanks to this wake-up call, should shut down her work now and find out how to get trained and licensed (if she is old enough, and if state law requires her to be older to get licensed, she should be thankful she found that out and stopped doing it before she got caught and shut down and fined).
She should also find out the law on paying taxes. Don't make assumptions about "I only made $X, it's not enough to pay taxes on." Pay up or live worrying that the state and IRS could come after back taxes. It the honest way, the grown-up way, to start a business career.
She would probably be a big asset to a professional nail salon one day and should pursue that if she loves doing it. Be ready to help her get trained (it'll cost) and licensed.
For yourself, get much more involved here. The post indicates you've been hands-off about her business but she needs firm guidance about stopping for now and doing research. I'd TELL her to return the money and then sit down with her and write out an action plan for researching all she needs to research.She will resist because she's made good money at something that's fun, but it's time for her to be an adult about it.