check the coolant temperature sensor wiring or transaxle wiring plug -
When this went wrong, did you notice that the engine fan was off? What you need to do is to get a good scan tool hooked on it and get the PID for the CTS and the TXtemp sensor looked at, if both are good then look at the PID for cooling fan state and test the fan too from the tool.
If the fan has died the lights will flash (on most cars) but the engine doesnt refuse to start. Is your car an ecostart car which shuts off the engine everytime it stops? - the starters on those cars die very quickly. It maybe a dead starter (self, salaff, sulf, --- whatever u wish to call it)