Angie, Where is the property located? I ask because your profile says you're in New York. The answer will be different in every state. Also, how long ago did you payoff the mortgage? You should already have your deed, as was provided to you by the title company when you purchased. But the lender might send you the original recorded mortgage, or they might send you a satisfaction of mortgage payoff. When you receive your satisfaction of mortgage, check to see if the lender already recorded the payoff. You may need to do that part yourself.
And yes, you can put the title in your daughter's name. Quit claim is the easiest way.
The deed is given to you after closing. I typically takes approximately 6 weeks to come back from the courthouse from being recorded. It does not work like a car where the bank holds the deed until it is paid off. You can put your daughter on title by doing a quit claim deed. I suggest contacting a title company or real estate attorney in the county in which the property is in.