First off let me say never use a brad or screw to fix this problem, it doesn't work and ruins the chair. Heat works, also applying vinegar will loosen yellow glue joints. If it was me the best fix is to turn a new spindle but that is not for most. Another option if you can't the tendon out is using a 2 part epoxy, you have a good surface area, apply the epoxy, position the spindle, wrap with wax paper. then I use rubber straps I make from bike inner tubes. Clamping tapered round surfaces is a pain in the butt. Epoxy can set in 5 minutes but I always let it cure for 24 hours. A razor blade will remove any excess epoxy.