Hi,
That's actually the install that doesn't work for me. (You'll note that it was released on 2012-10-22, but the cab files are all dated 2012-10-14, that's where I got my date from.)
I tried again though, and have a little more information for you.
I ran the level three uninstall as you suggested and then restarted the system.
I then ran the setup, and this time I got an error: The path c:\Pr4.00gram Files\HP contains an invalid character.
I clicked okay on that error and the setup program stopped.
Out of curiosity, I tried running setup.exe again... this time the error was: The system cannot find the file specified. And in the title of that window was ...\temp\7z5665p\Setup\hpzshlauto... (the first ... contained the full path to the temp file, the stuff after that was all I could see, the actual path and file was too wide for the window title)
Note that the computer I'm trying run this install on has been running Windows 8 since the fall of 2012. The printer only stopped working after I upgraded to Windows 8.1, approximately 1 month ago.