Cut five pieces of plywood to measure 18 inches square with the table saw.
Glue two pieces of plywood together at the edges to form a right angle, bracing the pieces with a chair to hold them in place. Let the glue dry for 1 hour. Pound in penny nails every 2 inches with the hammer to secure the connection.
Glue and nail the remaining pieces together in the same manner, to form a box shape with a top.
Cut the quilt batting into four pieces measuring 18 inches square. Layer the batting squares on top of the box. Cut a piece of fabric to measure 22 inches square. Stretch the fabric over the batting and attach it to the box sides with the staple gun, moving from the center of each side to the opposite toward the corners. Fold the corners neatly and staple them in place.
Cut the bullion fringe to the perimeter of the vanity stool. Attach the top fringe edge to the top edge of the stool with the hot glue gun.