Is that the light switch right beside the keyswitch?
Can we assume that the new engine wiring is recharging the battery? It would take a long while for a good battery to wear down, unless you had a load/drain other than cranking.
Pull off the plug and look at the terminals, or right beside them on the plastic, and you should see the wiring schematic/letters, S=starter, M=magneto, G=ground, etc..