I was trying to save you a little work. But, you are correct, it would be better if you entered the school where she was a full time student. The information goes on IRS form 8863.
Since you didn't get a 1098-T from that school, you gonna have to work thru the interview. I don't think you can say you got a 1098-T, when you didn't because it checks a box on form 8863.
You will need to enter the school info, including EIN.
On your return answer that you didn't get a 1098-T. Then answer that you qualify for an exception. You will still be asked for the tuition paid. Enter $4000. Do not enter any other numbers.
On her return, answer that you didn't get a 1098-T. Then answer that you qualify for an exception. Leave tuition blank. She will eventually be asked about scholarships. Enter $5117. Do not enter any other numbers.
But yes, check half time regardless of which school you enter.
