Flight Instructor reported an engine fire on the ground during engine start. A hand held fire extinguisher was deployed to extinguish the fire.

Narrative

I plugged in the aircraft at XA05 local and headed in to the school to meet my student; I instructed him to start the preflight and walked out just as he was completing. I started to spray the control surfaces with TKS fluid to remove the frost and disconnected preheat at about XA35 local. We completed the exterior and interior checklists and started the before engine start/engine start checklist. After priming 3 times we tried to start the engine and the engine did not start; we let it rest for about ten seconds and then tried to start again. After waiting for another 30 we tried to start to no avail. We primed again and worked the throttle and tried once more to start. After waiting again; we added one more shot of primer and worked the throttle cranked and the engine almost started. We gave it one last rest and tried to start and white smoke came out from the left side of the cowl; I switched off the fuel pump; pulled the mixture back and opened the throttle all the way and helped the Student turn off the fuel valve. I instructed him to continue cranking and grabbed the fire extinguisher and sprayed in to the cowl; there appeared to be more smoke so I asked him to help me open the top of the cowl so I could spray directly where the smoke was coming from. I depleted the extinguisher and the smoke subsided. Our A&P came out to the ramp and noted a broken fuel prime line and a chafed spark plug wire right net to each other and that the wire remained hot and appeared to be melted. Looking back; the only thing that may have changed the event is whether we continued to crank after having seen smoke