So a little update on this. I noticed that my Focusrite Scarlett 2i4 soundcard had a similar problem to the Helix, the firmware updater and ASIO control panel didn't work anymore.
I checked the Device Manager and found out that the Focusrite devices all had the event "Device not migrated". After trying to reinstall the drivers without success, I performed a system restore.
After that, I reinstalled the drivers and the soundcard was back to normal. The Helix though, no budge. Line 6 Updater can see the device and update it, so I updated it to the 2.82 firmware, but HX Edit still crashes when connecting to the Helix LT. I noticed that in Device Manager there is a line with Device not Migrated as well, so I suspect it is a similar issue to my soundcard.
I've seen the same thing in other threads here, unfortunately they were left unresolved: https://line6.com/support/topic/47763-hx-edit-no-device-connected-problem-help-please/
https://line6.com/support/topic/46308-hx-edit-crashing/
The telltale sign that something is very wrong is that HX Edit displays a transparent window even when the Helix is powered down and disconnected, instead of the normal black empty screen that I used to get before, when I didn't have Helix connected.
At this point I'm wondering if I should just reinstall Windows altogether, but I'd prefer to avoid it and see if there are any other options out there.