This suggests that the latest version has not been installed doesn't it?
In a word .. nope .. it doesn't mean that
should tell you which version you're running though (at the top of the long output)
and delete the HP printer if there is one.
Close the "Printers" dialog box.
Now see here:
Ignore all the command line pics .. start from the section "Using 'hp-setup' to configure your printer
in the first pic select "Network/Ethernet/Wireless network
" .. and click "Next"
Are there any printers discovered ? .. if so, select it and click "Next"
Give it a name such as "HP-5520" .. hopefully it will have automagically selected the driver (PPD) .. click "Add Printer"
again .. hopefully the new printer is listed there .. right click it and select "Properties" .. then click the "Print Test Page" button.