Take the window out of the door, and lay it on a flat surface. With the end of a flat-head screwdriver, pull out the end of the spline that's holding the screen in place. Pull the spline all the way out, and remove the screen.
Cut a new piece of screen about 1 inch larger than the opening. Lay the screen over the opening and center it.
Using the convex edge of the installation tool, push the screen on one side and the top into the groove. Pull tightly and push in the other side and bottom. Having someone help you hold the screen while you push it in the groove may help.
Put the new spline in with the concave edge of the tool. This will roll the spline right in for you. Make sure the screen is being held tightly and in place while you're putting in the spline.
Using a utility knife, cut the excess of the screen off right inside the groove on the outside edge of the spline. Put the window back in the door.