If you want to check Bonjour is working properly from your iPad's perspective on the network, there is an iPad/iPhone app available called "Discovery - Bonjour Browser" by Tidesoft. It's not free, unfortunately, but I can't see an alternative for iOS.
If you use it (or some other non-iPad based Bonjour tool for PC/Mac) then you're looking for an advertised IPP service (Internet Printing Protocol) from your HP printer. If you drill down into that, look at the capabilities list in the PDL record. It needs to include "image/urf". There should also be a URF record.
If none of that made any sense, I can give you a more step-by step run-through. The app is pretty automatic and straightforward to use, though.
If you
do see what I described on your iPad then AirPrint should work, and I'm at a loss to explain it!