After I installed iTunes 11, the following procedure worked:
1) Power-cycle my router and cable modem.
2) Reset the iPad network settings (General -> Reset -> Reset network settings).
3) [I suspect this is the step that really made the difference]:
a) Connect the iPad to the computer via USB.
b) Uncheck "Sync with this iPad over WiFi." Click "Apply."
c) Check "Sync with this iPad over Wifi." Click "Apply."
4) Restart Apple Mobile Service.
After I've done all of the above, iTunes is recognizing the iPad over WiFi, without any extra urging, for the first time since I've upgraded to iOS 6.0.