Remove the broken siding. Using a siding remover tool, gently pry the broken siding off the house. The top will be tucked into the siding strip above it, while the bottom will be nailed to the house through a siding nail strip. Remove the top part first with the tool. Next, use a hammer to gently pry the nails out of the house. Keep this nail strip, you will need it later.
Fix any torn insulation under the siding. Tape up with duct tape any broken patches of insulation if they are present.
Attach the new piece of siding. Hook the top of the new siding under the old. Nail in the bottom, using the nail strip and vinyl siding nails. The new piece should blend seamlessly with the old. Repeat this procedure until all the siding pieces have been removed and repaired.
Paint the siding to match. If there is a huge difference between the old and new siding, have your hardware store color match the color of the old siding with exterior latex paint. Apply this paint to the siding and let it dry. This should help make the difference less noticeable.