This is more fun than liars poker, but all good things come to an end.
A) The easiest and fastest solution 1) remove nipple, 2) remove as much powder as gravity will enable through nipple hole 3) replace nipple with Zerk fitting of same thread size 4) apply grease gun filled with cheapest grease available, 5) pump grease into barrel where combination of grease and/or air compressed in front of the grease will easily, safely and under controlled variables, expel the ball and attached implements. While pumping POINT THE BARREL into a 5 gal bucket filled with water as it might come out slowly, might come out less slowly.
B) A safe but not as easy solution for folks with no grease gun or zerk fittings, which works only because your broken rod is only a few inches from the muzzle ( PS, Ive done this and it worked like a charm): AFTER you have removed the nipple and as much powder as will come out with gravity and gentle shaking 1) Cut off the end the remaining long portion of your broken rod square, 2) using your broken rod as a guide, find or make what will be a 12-18 inch long ferrule of PVC , pex , bamboo cane or similar tubing that will slide snugly over the broken rod end in your barrel but the OD also slides easily into your barrel... leave it at least 12 inches long so you can have something to hold onto . If you cant get a good fit with what's on hand, you can bore out your tubing with a drill or sand down the OD to fit . When you have a ferrell the right fit, the ID will slide snug but not tight over your broken rod and OD slips easily down the barrel. You dont have to worry about the 6-7 inches sticking out beyond the muzzle. 3) Put a wrap of masking tape around the open end of your ferrell that will go into the muzzle..over lap the end of the ferrell 1/8" and then push that 1/8" flap onto and into the end 4) Hang your rifle securely in a muzzle-down position, but with a cord or chain so the muzzle is at least 24 " from the floor and it can swing out from vertical to horizontal position. 5) Tear off a 4-6 inch piece of Gorilla tape and gently attach one corner to your stock, as you will need to be able to reach it with one hand while holding the end of your muzzle with the other, 6) make a VERY LOOSE wad of dry tissue about an inch long and stick it in the open end of your ferrell that fits over your broken rod end and push it in about an inch 7) mix a 50 cent piece sized glob of quick dry epoxy ; 8) using a popsicle stick or any narrow old piece of flat wood or stiff carboard , hold your ferrell upright, fill the open end of your ferrell with enough epoxy that it the glue is about 1/2 inch short of the end of the ferrell, then prop it so it stays upright, glue at the top 8) pull away the masking tape from the upper end of the ferrell so there is no glue on the outside or end of your ferrell ,9) pulling your muzzle away from vertical as little as is necessary to see what you are doing, carefully slide your ferrell upward into the muzzle and JUST over the exposed end of your broken rod ..DO NOT PUSH YET. 10) Keeping your ferrell as close to parrallel with your bore as possible, SLOWLY push the ferrell upward and onto your broken rod, twisting as you push and thereby spreading the glue evenly over the end of the broken rod rather than extruding it into the bore and pushing the tissue down into the ferrell as you go, Push until you hit stop , then pull back about 1/2 inch. 11) Use your loose hanging piece of tape to secure your ferrell to the end of your barrell, thus keeping it in position. 12) put a pice of cardboard on the floor and leave rifle hanging with ferrell in place to let glue dry overnight. 13) the next day, when its dry, take rifle down - RESIST THE TEMPTATION TO TRY TO PULL THE BALL WITH THE FERRELL ! Instead, secure rifle in a vertical position, butt down, and gently TWIST the ferrell in a counterclockwise direction...you are (trying to unscrew the broken rod from your ball puller without pulling the ball puller out of the ball, but if that happens, it will make the next step unecessary) so that you can then reinsert your metal rod and twist it onto the threads of your ball puller , so try to get a sense of how close to center of the bore the threaded end of the ball puller is . It should turn easily and you'll know when it comes all the way off. 14) Position rifle to as vertical a barrel as possible, muzzle up, and insert metal rod ,using a hand at the muzzle to guide it toward the threaded end of the ball puller. It may take a few tries but in vertical position you will eventually get it over the threads and then thread rod end over the ball puller threads. 15) When metal rod is threaded onto ball puller, give it a few more clockwise twists to make sure ball puller is seated fully into ball and didnt come part-way out while removing broken wooden rod 16) Using your preferred technicque ( I favor securing the rifle in a vertical position and using a drop-hammer on my metal rod rather than securing my metal rod and using the weight of the rifle as a drop hammer), pull your ball and marvel over your handiwork.