You are correct that you should file a Form 709 for a year in which you give more than the current $15,000 gift tax exemption. (If you are married filing jointly, you can give $30,000.) BUT you are a bit out of date regarding the total lifetime gifts that you can make under the combined gift and estate tax. The exemption is currently $11.4 million! –except to your spouse, where you can make an unlimited gift. I hope that covers Your daughter’s problems.
Note, that you don’t have to pay taxes when you file this form. It’s just a way of keeping track of the gifts you make during your lifetime, and those you make after death — to make sure you don’t go over the exemption limit.