There are a good many options here. There is plenty of good music within the realm of church music--especially if you are inclined to believe in Christianity or you know of some well-executed Christian music and can listen to it for that alone. The witless starts whistling Kingdumb maladies, you start playing church music. Fair play.
Another good option is Christmas music. You might have a good library of Christmas music that you liked in December. The witless cleaning your windows is probably still trying to purge it from his head, and if he starts humming Kingdumb maladies, a good dose of Christmas music ought to be fair turnaround. Of course, you yourself may have heard all the Christmas songs you want to hear until next fall.
Still another option is rap and heavy metal. Led Zeppelin is excellent here, because the witlesses associate it with Satan. Don't worry if it's before your time--this stuff is timeless. Michael Jackson is another good option--the album Bad is perfect. Bonus: Bad has the song Man In the Mirror, which is perfect for any witless that might already be looking at self-improvement. Prince is good--so long as you stay away from his material recorded after 1999.
If you are in for rap, Ja Rule is excellent. From what I understand, it is common knowledge that Ja Rule is disfellowshipped, and most witlesses will get a headache if you have several CDs of Ja Rule going while they are trying to work. This makes the most sense if you start playing it the instant the witless starts playing, humming, singing, or whistling Kingdumb maladies. You might get good results from Eminem and 50 Cent, because those acts are downright filthy and the witlesses cannot stand them. Warning: If you have small children around, Eminem and 50 Cent might not be appropriate for them to listen to.
Or, you can play any bad songs. Any of the songs the witlesses bashed will do. If you were around when Hotel California was big, you are in luck because most witlesses cannot stand that album. The psychedelic rock bands are good--these were late 1960s, but many artists are timeless (I am thinking of the Doors). If you are into country, find those songs that glorify drunkenness and loose conduct. You might also have luck with Van Halen, Def Leppard, AC/DC, Judas Priest, and other late 1980s heavy metal that the witlesses object to. Guns & Roses and Poison were explicitly mentioned in the 4/15/1993 Washtowel in a study article as "bad". And, there are plenty of dirty R&B songs from the late 1980s and the 1990s you can listen to.
Finally, if your time allows and you are physically capable of doing so, just fire the witless window cleaner and do it yourself. It requires either a ladder or an extension pole, and you will have to learn the skills to get the windows clean. The good news is that it is very easy once you get a little practice, and you get to pick when. This works best if you own the place and the windows are accessible without specialized equipment. You can save plenty of money in the process, but you take on all the work (usually about an hour) and risk (you must learn to safely operate tall ladders if applicable) yourself. This will not work if you merely lease the place and the complex manager is the one that does the hiring and firing.