Writing a contract when the work is already done?

I suppose this is a bit of a weird question, but I couldn't find anything pertaining to my situation.

My wife works for a preschool and childcare, and she was put in charge of scheduling for the staff. In order to do that, she had to calculate the teacher-student ratio (it differs depending on a child's age) for each classroom. I was horrified to learn that this was all done with pencil and paper, so I wrote a web app to assist with this.

When the web app was complete -- and my wife had been using it for several weeks -- she presented it to the director of the facility. The director was impressed with it, and (some weeks later) had me attend a staff meeting to present it in proper form to the entire staff.

Now that the app is done, and has not been in need of any debugging or any kind of maintenance, really, the director informed my wife that she wants to pay me for the work. I certainly didn't keep track of the number of hours I put into it; it was a labor of love, literally: I love my wife and saw a way in which I could make her job a little less stressful. I can guess, but a guess is all it would be.

So as to my question. I have found plenty of templates for web development service agreements, but how do I write one when the work has already been done?