Okay "installed to" means after installation.
Are you telling me that after you run the patch program there is a new copy of EU2 in C:\My Documents\eu2_eng_v108. I don't think so I think that is where you dowloaded and saved the patch to. Which is not what was asked.
Okay one more time let's go through this.
1. remove all traces of EU2 from your system following the manual uninstall instructions.
2. Search for eu2.exe
3. If you find any, STOP and report the path to it here.
4. If you do not find any then run the patch installer.
5. Run the search for eu2.exe again, come back here and tell me the path if any that the search found.