The setup documentation for that HP printer is very weak, and I'd generally prefer to avoid needing any sort of add-on or set-up tool on OS X — something that HP seems dependent on here, at least this printer series. The P1600 series user manual is weak on details, too. In any case, go to page 53 and follow the instructions there to print a configuration page. According to the manual, somewhere on that configuration page is the current IP address for the printer.
Once you have that IP address, see if you can enter that IP address into Safari, and get to the HP web-based configuration interface. If you can get at that IP address, then you can view the configuration and settings for that printer.
In addition to access via your web browser, Apple has some configuration information available; use Spotlight search to launch Network Utility on your version of OS X — enter Network Utility into the search, and launch the app. If you can get to the web server in the printer from your web browser, then the ping command will almost certainly also work.
I can't tell if the printer uses DHCP — I'd expect it would — or if it supports setting an IP address manually — I'd hope that it would. I'd use a fixed ("static", "manual") IP address for your network. What address is used here and what address range your network is using depends on how your local network is configured.
Once you have the printer IP address, you can try removing existing printer definitions on OS X, and re-adding the printer. This via ?> System Preferences > Printers. You'll need the printer IP address to add the printer, once it's been removed.