The battery really shouldn't "wear down quickly". If you have too much voltage being dropped in the circuit it can give the impression that the battery is wearing down quickly.
A weak coil may not deliver enough spark when the starter is drawing a lot of current. I would get it good and warm, park it at home and then disconnect the power lead to the coil. Hook the coil up to a different source, like a six volt battery charger and try and start it. You may get quite a surprise.
