Wash the chair surface with a solution of 1 teaspoon of dishwashing liquid to 1 gallon of water to remove surface stains and dirt. Allow the chair to dry for 2 hours.
Separate the spindle from the top rail and the seat. Pry the wood from its seating, carefully breaking the glue bond and removing any screws or nails that hold it place. Gently work the pieces apart being careful not to break the wood any further.
Remove stuck spindles by drilling a hole into the bottom of the fitting, then tapping the drilled chamber with a chisel and tap to loosen the spindle.
Remove old, dried glue from the spindle and setting with a chisel, knife or paint scraper. Ensure wood is clean and free of debris.
Drill a narrow chamber in both ends of the broken spindle, taking care not to break off surrounding wood. The chamber should be narrow enough for a thin dowel, 1/4- to 3/8-inch diameter, to be inserted to support the rest of the spindle.
Insert a small piece of dowel into the drilled chamber. It should be the correct length to allow the broken pieces of the spindle to fit together neatly. Remove the dowel.
Apply wood glue to the dowel on both ends and secure into the drilled chamber.
Apply wood glue to the broken wood of the spindle and fit it together as it was originally.
Remove excess glue with a cloth.
Clamp the spindle and allow the glue to dry overnight.
Remove clamp and refinish the chair as needed.