Just something to try. Unhook both battery cables at the battery and touch the ends together for a few seconds. Then reconnect & see if it cleared any issues.
Any hardware store. Get a 1/4" X 1/2" bolt, nut & 2 washers. Pull the broken ends of the element close together and put both ends between the washers as you're tightening the bolt. Get it good & tight, be sure nothing will touch anything that could ground it. Put everything back together and...
A low pressure switch could keep contactor from snapping in. Or issue in general with low volts, transformer is place to start checking voltage, then wire to stay by looking for 24 volts there. Then back to unit circuit boards etc. Down here in south Ga lizards & green frogs crawl behind...