1989 Mazda 323

but the engine won't kick in,

If you mean its not cranking over-test the ignition switch/relay/park and neutral switch and the starter

Refusing to start while cranking over do below

Do you have a check engine light On or flashing prior to this problem?

If the engine's condition is in tip top shape-do below to determine if its fuel or spark problem that's if its cranking over and refuses to start

Get a carb cleaner and spray into the carb or the throttle body on an EFI. Did it start and die? If not disconnect a sparkplug wire or 2 and ground it to the engine -have helper crank engine over-do you have a snapping blue spark? If so-you have a fuel problem, check the fuel pressure to rule out the fuel pump/pressure regulator and listen to the injector/s are they pulsing or hook up a noid light. No snapping blue spark continue to troubleshoot the ignition system-power input to the coil/coil,distributor pick-up coil, ignition control module, cam and crank sensors-

If both fuel and spark is present-check the valve and ignition timing, this will lead you to problems with compression and valves opening and closing at the wrong time