I don't think not having a mortgage will hurt your daughter. Paying down your mortgage is often a strategy to help increase financial aid, because it reduces your liquid assets (cash), and your house is not counted as an asset to be used for college expenses (for public schools, may be considered for private schools).
From the FinAid! website...
"The Federal need analysis methodology does not consider the equity in the family's primary residence. So to maximize your eligibility for Federal aid, you could use your cash and other included assets to prepay part of your mortgage. Many private colleges and universities, however, do count your home as an asset when allocating institutional funds. If so, it may be worthwhile to get a home equity loan to provide funds for your children's education. Not only are the interest payments tax deductible, but the loan reduces your assets."
For more tips to help maximize your financial aid eligibility, visit:
http://www.finaid.org/fafsa/maximize.phtml