Take the damaged rung to a home improvement store and purchase a new wooden dowel in the same size. Cut the dowel to the same length as the old rung and stain it to match the chair if necessary.
Scrape out the rung holes with a screwdriver; this will remove any debris or old glue residue, making a clean space for the new rung to slide into.
Fit the dowel into the holes without any glue. Use an electric drill with a 5/64-inch diameter drill bit to make a very small hole through the leg of the chair and the end of the dowel, at a right angle to the rung hole; repeat this on both sides.
Remove the dowel from the rung holes and apply a thin layer of wood glue to both ends of the dowel, as well as into the holes. Fit the dowel back into the rung holes; the small holes that you drilled previously will allow excess glue to escape, ensuring a tight fit.
Wipe away any excess wood glue that oozed from the holes, then hammer a small finishing nail into each of the drilled holes to assist the glue in holding the dowel in place.