first lesson is that the same species of wood will stain the same with the same type of stain used.
ie, pine and oak will stain differently with the same stain.
Make sure your woods are the same.
second, this will have to be a trial situation only. Find a stain that looks the same as the old and then wipe on until the color matches the rest of the chair. You might want to feather the stain over the other areas to blend in some. Then seal with a good sealer.
Stain in the direction of the grain. Sand in the direction of the grain. Many glues do not stain and will leave a bleached out area. Many fillers will not stain and will leave the same.