You will need 2 dowels: one the size of the broken spindle and a smaller one
cut off the broken spindle so that you can drill out a hole large enough for the smaller one to be inserted inside of it - this would be the lower portion of the spindle. In other words, a dowel within a spindle.
Now cut you replacement spindle and perform the same operation on it - this would be the top of the spindle which attaches to the arm.
Glue the dowel into the spindle & the two spindle pieces together & insert.
Remove the entire broken spindle and reglue the broken pieces together. Drill a hole deep enough for a smaller dowel to be inserted into the spindle down past the broken portion, glue everything and reinset.
Remove the broken spindle, find, make a new spindle, insert.
Hope this helps.