The Kawasaki manual says to clean the air filter using a low flash point solvent (like kerosene) and apply a light coat of clean engine oil on the filter.
I thought the "sticky" No-Toil oil might be better at trapping dirt than engine oil so I used it on a new filter.
After I thought about it a little more I kind of wondered what sucking some of that sticky stuff into the carbs would do to them so I decided to clean the No-Toil off the filter and go back to using regular engine oil. I used a bucket of warm water with some Tide and then a little Dawn dish washing detergent on the filter.
The filter, which is about a month old, disintegrated. The fibers which were attached to the foam started to come off - so the filter is ruined. I'd cleaned the old filter a couple of times with kerosene and re-oiled it with engine oil and never had a problem.
I'm ordering a new filter and will be throwing the No-Toil and the ruined filter in the trash.
Luckily I'd kept the old air filter which I'd cleaned before I put it away so I have something for the bike until the new air filter shows up next week.
I don't know what No-Toil is made of - it is sticky and a bit hard to get off your hands or anything else it gets on. There's some special cleaner you are supposed to buy to clean it off a filter and they sell another cleaner to clean it out of the air box - they don't sell anything to get it out of the carbs though.
Engine oil and kerosene have multiple uses, are easy to come by and work just fine. If the engine happens to suck some engine oil through the filter into the carbs it will be dissolved by gasoline flowing through the carbs. Also you can clean and re-oil this particular type of filter multiple times using kerosene and engine oil rather than having it fall apart the first time you try to clean it after using No-Toil.
I sent an email to No-Toil customer service to see if they will give me money for a new filter - but I'm not holding my breath on that.