Hi Mike:
When you have the mower in a state where the engine won't start, first do a spark check. This will rule-out any spark-related issue (or confirm it). It is rare for a coil to fail, but the cap, wire, plug etc. should be inspected to be sure of no other spark-related issues.
So, assuming you have good spark, the we're probably looking at incorrect air/fuel delivery.
1. If any doubt at all of the condition of the air cleaner element, replace with a new one. A dirty or partially clogged air filter will cause difficult or impossible starting conditions.
2. Drain out all the old fuel from the tank, fuel lines and carburetor bowl. Fill with fresh gasoline only, and make sure the spark plug is clean, dry and gapped correctly. Confirm the choke plate is fully closed on the cold engine. and opens fully after the engine has been running for at least 2-3 minutes.
3. Some versions of your model (you have a "K0" version) had a problem with sticking autochokes. If you'll get the serial number off the back of the mower deck, I can see if you mower is in the affected range, as well as check on the warranty status at the same time.
4. If the autochoke is not the issue, it likely is some debris or dirt partially clogging the carburetor. It can be easier/faster to simply replace the carb vs. trying to properly and correctly cleaning one 100% For example list price on a whole new carb is only $15.91 list price.
Let's get that serial number and go from there: