How to repair broken off wood from a chair?
An easy way to test out stains is to apply an ample amount of the filler you are using to a scrap piece of wood. It is unlikely you will be able to match the wood. But judging by the amount of filler you have, I assume matching the filer is the biggest concern. Sand the filler on the chair and the scrap using the same tools( sandpaper etc ). Then test out the stains you pick on the scrap piece and hold it up to your chair. This way, you don't have to commit to a stain until you are fairly sure it will match. 

Also, sometimes a paint supplier, Porter, Sherwin Williams, etc. has some professionals who actually still mix tints/stains. They can usually do a decent job of getting close to the stain match. Bring a good picture of the chair stain and the sample piece you made and they may be able to help. Otherwise, just get a bunch of small stain cans from your local big box and test away!

Happy hunting!!!