I don’t think that log helps. Anyhow: /var/log/kdm.log
You can try login to a virtual console:
Platform-specific Desktop Tips
Then try to fix the broken dependencies.
sudo apt-get update
sudo apt-get install non-qubes-whonix-gateway
It is probably a lot easier to just install a new Whonix 13 Whonix-Gateway rather than trying to fix a broken Whonix 12 → Whonix 13 upgrade.