Hi I'll try and help you out.
Do you have a compression gauge? This would be something worth checking.
Does this have a heater or a wrap or is it someplace where it can be warmed up?
If the compression is good, then I would check the flywheel key. The key controls your ignition timing preset and it's not at all unusual for one to shear and cause the problem you're experiencing.
The fact that it won't hit on ether tells us to look past the fuel system for the time being and look at the other two systems necessary for combustion, compression and ignition. A compression check will give an idea as to the sealing ability of the engine, the rings and valves. Checking the flywheel key will verify that the ignition which you've already checked is being delivered at the right time.
This is where I would start in the shop with the symptoms you describe.
If the engine pops loudly through the exhaust or wheezes back through the carburetor that might indicate a problem with a valve, but generally on a twin it will only be on one cylinder. It's possible that the cold is preventing good sealing by the rings. 950 hours isn't an awful lot of hours on a quality engine like a Kawasaki, but there could be enough wear to loosen things up a little with the cold. But again, it's a twin so you would expect only one cylinder to act up. If the flywheel key has sheared, that will affect the timing to both cylinders. So if you feel that neither cylinder is performing, the key would be the first thing to check.
Let me know if you can check these things and what you get. Thanks, PK.
There's no real problems to look for that I can think of. You are at the point where a 300 hr service should be done, if it's been done regularly. This would include pulling the heads and cleaning the carbon off and I believe Kawasaki suggests lapping the valves every 300 hours. Since you have a problem with the cam gear you might take the opportunity to go ahead and freshen the engine up with rings and a cylinder hone, but that's entirely optional.
As for the starter, I'd pull it off and check it out. It may have a worn gear or the pinion clutch might be wearing out.
I haven't had a water-cooled Kawasaki in the shop yet. I've owned a number of the air-cooled engines and had them on commercial mowers and really loved them. Good power and I don't think I ever had a serious problem with one.