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You can pay off her loan balance and you would not be taxed for doing so. By making the payment for her you would be making a gift and that gift would be more than the annual allowed amount ($14,000) but as long as you have not used up your lifetime allowance for gifts ($5.43 million) you would not pay tax.
You would need to report the gift (form 709) but on that form you claim the first $14000 under annual allowance and then the remaining under Unified Credit (lifetime allowance) leaving you no Gift tax to pay.
Please advise if you need more information or rating in a positive way is acceptable.
Your daughter is not taxed on the gift, neither does she even have to report.