Not starting hot is a funny thing to diagnose.
Usually it is a mal adjusted carb or a sticky valve.
After that you have a magneto coil on it's way out but I would expect it suddenly stop during mowing, or at least miss when hot.
Valve adjustment is an easy job you can do at home and takes about 1/2 hour or so depending upon how much cleaning has to be done.
The valve lash should be checked every 200 hours or so which would be every 2nd or 3rd season and go hand in hand with pulling the blower housing & cleaning the fins.
Unless the engine was turning over slowly it is not likely to be the starter, starters rarely give any problems till the drive gear has worn thin.
I have replaced 1 starter in 700 services and 2 sets of brushes and all of these were on mowers that were left in the weather.
If you want to have a go yourself then live with it for now and and pull it down at the end of the season.
Kawasaki manuals are all here
http://www.mymowerparts.com/about_us.php and the only part you will need to but is a pair of rocker cover gaskets .
BEst done before you change the oil as the gaskets are generally stuck fast and bits of gasket always end up in the engine.