Server 2012 R2 System Disk Full (Unknown File)

This is because you're not doing backups. Your log files (.LDF) get cleared when you do a backup of the database. You can either do a real backup and shrink the logs, or you can use simple recovery mode and shrink the database files to trick SQL Server into thinking that you did a proper backup, but all it will do is free up space. It will prevent your ability to replay the transaction logs and roll back. Be sure you understand the implications before doing this.

Of course, this may fail given your extreme disk space issue, so you may need to extend the drive first (which you said may not be easy) before any of these commands can work.

As I read this 3 obvious causes come to mind:

1. root-kit: You're best bet is to boot from clean media (like a Linux live-cd) and scan the disk from there using one of the online virusscanners. Even if the scanner doesn't see anything suspicious you might see some very large files or weirdly named folders on the disk that would otherwise be invisible from the infected system.
2. Some sort of disk-corruption that chkdsk misses. Did you ran chkdsk during boot instead of on the online file-system already ? If not try that first. (Run chkdsk c: /X from a cmd prompt to force it into a boot-time chkdsk.) If that is clean too a disk-corruption issue seems unlikely.
3. Some sort of hidden file (like an ADS stream) that got completely out of hand. Try clearing the Internet Explorer cache. This is full of ADS streams that are completely invisible and sometimes get 'run away'. There is a tools to hunt ADS files down in the SysInternals Suite. You can give that a try too.