The Generation 1 (pre-2008) KLRs have CDI ignition, unlike the latter-day models.
For spark on a Generation 1, you must have: Alternating current from the exciter coils of the stator feeding the CDI unit, recurring pulses from the pickup coil triggered by the timing lump on the rotor (flywheel), and an operational ignition coil.
You can measure exciter coil AC when the engine is rotated (remove the spark plug and grind it with the starter). You can monitor the pulses from the pickup coil. You can check resistances on the CDI terminals, and the ignition coil.
You can check connections for continuity/short-circuit; the kill mechanism is shorting the exciter coil voltage to the CDI by kill switch or by ignition switch in the OFF position.
Good luck; Google "KLR650 wiring diagram) for a running start at your problem.