vineyridge,
I realise I am 2 years late to the party. I also realise you have been trying to connect the printer via USB.
Still, I just made my Envy 6400 series work on 5G WiFi, on Slackware-15.0. The steps can be replicated on any distro, so I thought I'd share them, maybe yourself or someone else will find them useful.
The connection part:
1) Follow the printer's instructions to connect to WiFi 5G -- yes, for this part you do need to read the damn manual.
But there's a gotcha here. In your router, make sure Wireless mode is set to "802.11a/n/ac mixed". If you prefer "802.11ac" because you want things tidy, then you'd be out of luck (don't waste time figuring it out

).
The scanner part:
2) On WiFi, we need the sane-airscan package. It has one dependency: avahi, which itself has one dependency: libdaemon. Installing these packages on Mint might be straightforward because it has a graphical package manager.
3) As root, edit your /etc/sane.d/dll.conf and make sure you have the escl backend listed and no # in front of it. For faster scanner initialisation, you could put a # in front of every other line, so that sane doesn't try them all during startup.
4) As normal user, in a terminal, type scanimage -L. If you see your device listed then you are done with the scanner part. But do make a note of your printer/scanner's IP address, we'll need it in step 6).
The printer part:
5) We need at least hplip-3.21.4 (that's the version in which HP included the Envy 6400 series). I use the latest hplip-3.22.4, which has avahi as a dependency, but we already got it from step 2).
6) As normal user, on the command line, type hp-setup <IP-Address from step 4)>. See your device listed? Good, go through the works.
7) Done. Watch cups and sane work their magic. Even the document feeder works -- oh yeah!
Again, hope these instructions are useful to someone.
kdato