So, regarding my cry for help...
I did get several replies and did more research on my own. The TL;DR up to now is "I have a fully functioning device with no input method and my data may well die on it":
- The device is passphrase-protected and encrypted so I can't simply connect an USB cable and use MTP.
- I can't connect a mouse or keyboard as LG, in their endless wisdom, didn't design the USB port with enough power in mind so it can't support USB OTG on its own.
- Google then removed USB OTG support from the Nexus 4's kernel. It's not as if powered USB hubs existed so this is obviously the correct path of action.
- While I can install new programs via Google Play, Android 4.0 and above prevents newly installed programs to start without user interaction.
- LG points towards a third-party service for out-of-warranty repairs and as part of their Terms of Service, you have to forfeit all data as they "always update the software", i.e. they will prolly ship random other devices to you on a regular basis instead of what you sent in.
- The Nexus 4 is running stock Android, locked bootloader and all
The last two options I see are
- Try to find a way to get a custom ROM onto the device with the help of USB cable and physical buttons only without destroying the encrypted data (yeah, right...)
- Try and source a display so I can repair the device myself. But as not even ifixit.com offers a howto or parts... I suspect this may fail.
And I can not even be reached under my normal number as I don't dare turning the device off and/or removing the SIM as that may prevent me from recovering with the running device, somehow.