if you have broken cam gear, "usually" (but not always) the compression pressure will be off. I've had them show up a tooth off and still make ~200 psi. Pull the "pan", gear is missing a tooth or two.
that engine has an issue. The carb sits down between the cylinders and directly above the muffler. Gasoline has two different blends for the seasons. Winter blend, and summer blend. Winter blend fuel is blended such that it vaporizes at a much lower temp than summer blend fuel. Winter fuel can boil at temps as low as 80 deg F. So the carb sits there where it's at, you shut it off, the muffler is hot (obviously), engine is hot, fuel boils out of the bowl and the next time you jump on the mower to start it, it's hard to start. You have to crank and crank until the pump can refill the bowl. Remember, gas is made for cars now, and none of it (not even "no ethanol") is made for lawn mowers or any other small engine.
aged fuel, no matter how much sta-bil you put in it, sucks. You get maybe a month or so out of it with sta-bil, and beyond that, throw it out or put it in the car. No good on anything with a carburetor.
make sure the choke is adjusted such that it is actually closing completely. If not, it WILL be hard to start. The choke creates a vacuum through the carburetor which helps pull fuel through the main and pilot circuit. At cranking speed there is not enough airflow generated by the engine to pull sufficient fuel, so the choke MUST close all the way to supplement that vacuum.
Kawasaki changed the fuel pumps up, the new ones are more compatible with E-10 fuel. I like to suggest replacing them even if it works. Also the new pumps on a lot of them now have a separate drain line rather than just a "weep hole". The weep hole was, again, right over the top of the muffler so if it ever leaks, and they do, it's a great way to burn a mower to the ground. New pump has a nipple that you put a hose on and run it down the side of the frame, away from the muffler.