Does Kawasaki FC540V sound right?
First off to verify a no spark issue, I usually remove the spark plug and shoot either carb spray or starting fluid in the plug hole, just a short burst will do. reinstall the spark plugs and plug lead, try to start if it fires and dies, probably not a spark issue. if it doesn't fire, no spark. Next what i do is remover the plastic or metal engine cover (blower housing) and locate the ignition coil... there should be a black kill wire hooked to it, unplug it, and repeat the steps aformentioned. If it fires with the kill wire unhooked, something is causing the kill wire to short to ground.
Now, IF it is a FC540, the ignition system contains 2 parts the coil mounted next to the flywheel, and a igniter, which is basically a electronic points set, that tells the coil when to fire... It is possible the igniter is bad.
Also another variable is the engine's spec number, FC540V-xxxx, looking at the parts illustrations, some models of the FC540 Used an igniter, and some used an integrated igniter, which is located inside the ignition coil and is replaced as a unit.
EDIT: if this is the same JD 180 not an L180, this thread may cover the solenoid issue.
https://www.lawnmowerforum.com/threads/john-deere-180-starting-issues.25790/
not aware if there is a procedure to test an igniter, one of the true tech's will probably need to answer that.