Loose chair rungs are a fairly common problem but the solution can be simple. These joints (usually dowels) can be completely repaired. Begin by cleaning the old dried glue and any debris from the dowel hole. You may also want to lightly sand the dowel itself. Dry fit the dowel into the hole and drill a tiny hole through the leg and the rung from a different angle. Separate the joint and add glue to the dowel and the hole. When the joint is put back together, the hole you’ve drilled will allow excess glue to escape and allow you to drive in a finish nail holding the joint in place.