We pay for you daughters higher education tuition and expenses, but she does not qualify as our dependent, what can we do?

Only the person who claims her as a dependent can claim her educational expenses for the tuition credit.  If she is no one's dependent, then she can claim the tuition credit on her own return, even though somebody else (you) made the actual payments.

Be advised that there are restrictions on unmarried, full time student, under age 24, getting the refundable portion of the main tuition credit.  It is usually better, if the parents claim that credit.

So, if she is a "college age" student, why do you think she is not your dependent?

There are two types of dependents, "Qualifying Children"(QC) and Other ("Qualifying Relative" in IRS parlance even though they don't have to actually be related). There is no income limit for a QC but there is an age limit, student status, a relationship test and residence test. The support test, for a QC, is only that the child didn't provide more than half his own support. The support test for a Qualifying Relative is that the taxpayer provided more than half the relative's support.

If your daughter is not your tax dependent, then you cannot claim her education credit.

Only she can claim it on her own tax return.

Please read this IRS document for more information.