Hooray! I have SQL 2014 installed. I looked in the Windows Registry for any keys that referred to old versions of SQL Server here: HKEY_LOCAL_MACHINE\SOFTWARE\Microsoft\Windows\CurrentVersion\Uninstall and deleted them. I also moved any data files to a new directory with a different name. Before this I had used the Remove Programs option of Control panel to delete any old SQL Server references. I didn't think this would be enough but I got lucky. The only reason I could be pretty reckless in deleting things is: 12) I backed up the Windows Registry before I started hacking it and 2) the server doesn't have anything that is being used right now.

Still, I had to get this working without any help from the "Network Admin" so I'm really relieved it worked.