Please read the article linked below. The gift is over the annual limit, so you would need to file Form 709.
To claim herself, she would not be your dependent, therefore she would need to have provided her own health coverage.
There is another option which is no one claims the student. The student can then get the "Non-refundable" portion of the education credit.
This is done by the student selecting "Yes' to "Someone else can claim me" but "No" to "Someone else will claim me"
You also would not claim the student. There is no exemption amount this year, so you don't lose that, but you might lose on a different credit.
https://www.thetaxadviser.com/issues/2010/aug/nichols-aug-2010.html
CLICK
HERE for IRS Pub 970 Tax Benefits for Education