LAN connection problems…

I agree with michaelkenward that this might a a complicated problem.

Apple Airprint.  It is not entirely clear (to me) whether Apple Airprint uses

• A direct WiFi connection between the Apple device and the printer (in which case knowing the network IP address of the printer is of no use), or
• Searches the local network for a printer that supports Airprint.  ("Air" would seem to argue against this).

There is a list of printers compatible with Airprint:

https://support.apple.com/en-us/HT201311 

After verifying that the HP printer is on the list, it might be worth a few minutes to look at the printer user guide to see how HP describes using Airprint.

This is also a case where my primitive understanding of iPhones is not helpful.  My Android devices, for example, have tons of apps which can be used for network troubleshooting.  For example, I can 'ping' the IP address of specific devices and see if  they respond.  ("Out of the box" my Epson printer (a) does respond to ICMP ('ping') and (b) has a web page that can be accessed from the network.  Thus, my network apps can 'ping' it and I can open the web page using a web browser on my phone or tablet.  "Out of the box", Windows computers are set to not respond to ICMP and the Windows firewall has to be told to allow it.)

Does the iPhone have similar apps?

Airplay from a iPad to a Libratone is another topic which displays my colossal ignorance.  Does Airplay (a) use the network for connection, or (b) connect directly to a device?