The carb kit is around the $ 10 mark , kawasaki parts are a bit on the expensive side, just a volume thing, too few parts going through the warehouse add a lot of fixed costs onto everything.
So clean the carb & rebuild it.
Before you start take lots of photos and in particular where springs & rods go, some line to mark cables at the clamp so they go back in the same place.
You will need to do a double oil change at least.
I usually suggest getting a gallon bottle of cheap oil from the supermarket.
Drain out all of the oil , tilt the mower if you have to and leave it draining while you do the carb.
Refill with the cheap oil start the engine, run it till warm ~ 10 minutes then change the oil again .
If you have enough oil repeat till you run out then change to your regular mower oil and replace the oil filter.
Change the filter again at the next oil change then back to scheduled servicing.
Worth buying an in line fuel tap , get it from a mower store cause there are a lot out there with the wrong seals in them that go to mush when exposed to modern fuels.
Fit the tap as the first thing in the fuel line from the tank so in future you will be able to shut off the fuel to replace the filter.
In theory should not be needed.
In theory you don't need house insurance either.
Turn the fuel off when you have finished mowing , let the engine run out THEN TURN IT OFF & REMOVE THE KEY.
After a while you will know exactly how much fuel is needed to get back into the shed so it will run out almost as soon as you get there.
More insurance is to get a 3" round key ring and fit the mower key to it.
Remove the key when the engine stops and hang it over the dip stick.
Thus you will remember to check the oil when ever you go to use the oil
I write "check oil before starting" on the blower housing for serial offenders and it seems to be working.