This is basically it, but I’d like to add something. Don’t bother drilling through the long piece. Instead, find a peg of equal diameter to the one that broke, cut it to size (+some that will go in the balls) cut off the old peg, drill holes in both balls, and superglue in the new peg.
You can get brass rods in scale modeling hobby stores and/or online and they’ll be crazy strong. Alternatively you can get plastic rods in a variety of diameters.