As you're hunting for parts for this, it was a model 22R, Kawasaki used three carbs through the years, their ID numbers were 12497, 12702, & 32113
all carbs used were superseded to Deere part # AET10909 (32113) usually the number was on an outer edge of one of the mounting flanges, but I have seen them on the webbing running between the flanges also. Your biggest issue is all the choices are obsolete, but if you look it up at JD parts. com, some pieces are still available. Just depends on what you need. This mower was introduced as Deere was just beginning to get into golf course equipment in the late 80's - early 90's you might find a golf course equipment salvage place that has one, but I'd guess you'd end up with one as bad as yours