Only Cub Cadets - No Spark

I recently brought home a complete 128 that had not been started in years. It came without a battery. We put it up on the lift and did very basic stuff to see if it would start. New gas, cleaned bowl, cleaned spark plug. My son put a battery in while I cleaned the carb a little. Time came to turn the engine over and check for spark. It cranked over great and appeared to have good compression. However, the spark was not consistent. I installed the plug and cranked it over trying to start it without success. Since I didn't want to run down the battery, I told my son to put on a battery charger. That is when I noticed he had put the battery in wrong, positive to negative. After having a good chuckle we hooked it up correctly and the engine started very quickly, running smooth and strong, no smoke or knocks. It died after about 30 seconds and would not start back up. Now it has no spark. I am guessing the coil or condenser has been messed up. Any Thoughts on the best way of finding out? Electrical problems are my biggest challenge.