While a recent late payment will damage your credit score, the effect of the late pay will diminish with time, as long as you make it a point to pay the rest of your bills by the due date. Having a recent perfect pay history can begin to overtake the effects of any late payments you may have had in the past. Your payment history makes up 35% of your credit score, so paying on time has a large influence on your credit score.
So if you want -- and I'm not recommending this; I'm just saying it's a strategy you might decide to use -- you can dispute information that you think is accurate in hopes that the creditor will not respond. (This is the strategy many credit repair firms use to try to improve their clients' scores.) If the creditor doesn't respond, the entry will get removed.
That's very commendable of you to handle your daughter's financial problems that way. I used to be employed as a loan officer in finance, but things have changed so much in the last 20-30 years. I accomplished something very similar to her situation, but I started in the fair range on scoring. I raised mine 204 points in less than 9 months. Thanks for passing along this great advice and experience.