Hi, I'm Robert and I'll be helping you with your printer question.
The problem may lay with your router if it wont support bonjour which hp likes to use for all apple products. That seems to be the direction apple is going as well so the new iphone may depend on it while the old one didn't need it.
what type of message do you get when you try to print from the iphone ? Does it find the printer at all ?
The message I get on the iPhone is that when I select the Print function, it can't find any Airprint printers, This is the same message I got with the previous iPad. The new iPad is a model 2, which is the same as the previous one. The main difference between them is that the new one has iOS version 6.1.3, which is the same as the iPhone. Both are upgraded from iOS vers. 5. I gave up trying to use Airprint last year when I installed the HP e-Print app, and only came across it yesterday by accident. HP e-Print requires that I SEND files in email to my printer's email address for printing, but it truncates the data and reduces the size of it significantly. It has limited usefulness but it was the best I could do. I see now that Airprint prints the same image on the full sheet, and has better quality, and prints directly, and much more quickly than having to send it by email. I don't care about HP e-print if I can get Airprint working on the iPhone. It seems to work very well on the new iPad, I just don't know why, and what I have to do to get it to work on the iPhone. I believe the iPad is the same as the old one, including the settings, which were restored from the old device. Only the iOS has changed from vers 5 to vers. 6.
Of further note: my understanding is that the newer systems DON'T require Bonjour at all; also I have recently switched from an older AT&T DSL modem and a Belkin wireless router for wireless and internet, to AT&T's much faster U-verse High speed Internet wireless gateway for all my wireless and internet access. Does any of this help? Thank you, Kurt.