Remove the broken dowel. Unscrew the broken dowel if screws attach it to the chair. If the dowel does not come loose, use a drill to drill it out or a claw hammer to pry it from its hole. Saw off the broken dowel if necessary.
Clean residual glue left from the removed dowel. Use rubbing alcohol to soften the adhesive and then use a damp cloth to wipe away any glue that remains on the chair. A mixture of water and vinegar is also effective for removing glue from wood.
Apply a small drop of glue to both ends of the dowel. Also apply a small drop of glue directly to the location of the chair where you will place the dowel.
Place the new dowel into the spot from which you removed the broken dowel.
Secure the dowel in place. Place the dowel between the clamp's jaws and crank the clamp until it is tight against the wood. It will be tight when you can no longer crank the clamp.
Paint the chair with a wood primer to create an even color. Add a clear coat or base coat if the chair's appearance requires. When repairing chairs that already had a coat of paint on them, apply a new coat of paint. When repairing chairs that had a wood finish on them, apply a similar finish.