iOS printing without AirPrint
Printer Pro won't solve the problem because


  Printer Pro is the application that lets you wirelessly print from the iPhone. It can print directly to many Wi-Fi printers and to any printer attached to your Mac or PC via helper application installed on your computer.


If you search for "HP print" in the App Store, there are quite a lot of hits. HP All-in-One Printer Remote looks kind of helpful (but I can't test it myself).