Here's how I did it:
I stripped off the fabric and the trim and used it as a template to cut out my new fabric:
I used spray adhesive to glue down the top part over the foam, doing a little bit at a time to make sure the fabric was smooth and taut.
I wanted two strips of self welting- one along the top, and one along the bottom. To make the self welt, I joined strips of fabric cut on the bias. I put the self welt cording in the middle of the fabric strip and sewed the cord snuggly into the piece of fabric.
I used iron on hem tape to adhere the two strips of welting to the fabric that goes around the side of the footstool, and then more spray adhesive to glue the entire piece around the footstool.
The seam turned out a little messy, so I used some nailhead strips to create a decorative column along the seam. I put 3 more columns so the footstool would look balanced.
The project took me all weekend. Partly because the glue had to dry. And partly because I didn't really know what I was doing and had to figure it out as I went along. All in all, I'm pretty satisfied.