I'm still not clear on what the cancellation policy was with the university. However, they have to honor the policy they stated. If they said that you can cancel by July 27th and you would only receive a charge for room and board for a week and a half, then they need to honor that cancellation.
Universities are notorious for their bad administration. You will need to contact the university housing administration by phone, and try to get a manager or supervisor on the phone to explain the situation. The most important part will be getting past the initial customer service representative that answers the phone. They usually do not want to help, or do not know how to help. That is going to be the quickest way to resolve the issue.
If that does not resolve the issue, you need to dispute the charge with your credit card company, stating that the charge was not the amount you authorized to be charged (if it is on a credit card). If it is payment by an invoice, do not pay it.
Then you need to send a written letter to the housing administration at the university which explains that the University's policy on cancellation allowed you to cancel by July 27. You call and cancelled and also wrote a letter on July 27 (do you have proof of this? If so include it). Then you need to point out where that cancellation policy is. Finally, you need to demand that they reverse the charge without any penalty to your daughter.
You need to handle this quickly so there won't be any strange result like their system withdrawing your daughter for failure to pay fees or something.
If all that fails, then you have to sue in court for a Declaratory Judgment
. However, you are a long way from that point and should be able to get it worked out with the University by being persistent and clear in your explanation.