The normal way that a distribution from a 529 account happens is because the account owner made a request for a specific amount to be distributed to either the account owner, the account beneficiary, or a school. The most likely reason that the 529 administrator sent $16,571 to the school is because that's what you requested. How else would the plan administrator know how much you wanted to withdraw?
If you are going to count $4,000 of the 529 distribution as a non-qualified distribution so that you can claim the AOTC on your tax return, then you need to designate 39.04% of that amount (or $1,561.52) as income taxable to your daughter. This is the same earnings/distribution ratio as the overall $16,571 distribution.
You do have an amount to use: the Cost of Attendance (COA) that the school is required to make available that will have the school's estimate of room and board costs. Some schools only have one set of COA numbers for room and board, and some schools will have different sets depending on the student's living situation: on campus, off campus away from home, or at home. Find the appropriate COA numbers provided by your daughter's school. For textbooks and other required class materials, does your daughter have no records (receipts, order forms, etc.) for any of these purchases? If not, these records must be saved going forward if you want to use 529 funds to purchase these items.
This is your biggest problem. Sending the 529 money directly to the school, in an amount that almost exactly covers the tuition expense, makes it really hard to claim that the money was actually used to pay for qualified room and board expenses or other qualified expenses, like textbooks or a computer. If you had the 529 distribution sent to your daughter, then you could have decided how and in what amounts the money was spent on the whole range of 529 qualified expenses. You could have paid $4,000 of the tuition from other (non-tax advantaged) money, claimed the full AOTC on your taxes, spent the remaining $4,000 of the $16,571 529 distribution on textbooks and living expenses (up to the school's COA for room and board), and there would be no tax implications.
Yes, it is reported on your daughter's tax return. There is no 10% penalty because of the penalty exception that applies when 529 money used for a qualified expense is deemed taxable so that an education tax credit can be claimed. (See exception #5 under "Additional Tax on Taxable Distributions" on page 60 of IRS Pub. 970.)
The education tax credit may be claimed on your return, but the earnings portion of the $4,000 non-qualified 529 distribution is taxable income to your daughter, because the distribution was made payable to your daughter's school (and reported under her SSN). The same would be true if the distribution was made payable directly to your daughter. It would only be a tax liability for you if the distribution was made payable to you, the account owner (and it would then be reported under your SSN).
