Cam chain / timing chain tensioner failure is common on the 600. So is chain wear, increasing the possibility for the engine to jump time. And bend valves.
At this point, I'd be pulling the tensioner out and checking that it is all intact and working. They are prone to fail, and many of the older ones were cobbled to function as manual adjusters, rather than spring tensioned adjusters. Your's may have been replaced with an after market component, and require manual adjustment. Or be shot and need replacement. i contend this is where frequent oil changes are evident. Oil that has "broke down", no longer offering the needed protection it was designed to provide, will allow excessive wear in the timing / cam drive system. The Mojave ATV's are notorious for cam system issues, which uses a similarly designed engine. Not all that unusual for a low mile engine to need a replacement timing chain, if it has not been well maintained.
On the flip side, someone will pipe up and claim they only change oil every 10,000 miles, replacing with used oil from the county hazmat recycling tank, and they never had a timing problem, so that couldn't be it.
One step at a time. Check your tensioner.
BTW, if you need anything for that bike, there is a complete, running when stripped, KLR 600 being parted out on EBAY right now. Email seller for additional information on other parts he has.