I'm going under the assumption that your eq files are correctly in the location you specified and you did not miss a folder/do not have a folder in that.
The only thing I can come up with for 10 seconds of thinking is to remove the quotation marks around the path, so that it is just:
C:\Program Files\Sony\Everquest\eqgame.exe patchme
not sure if that will work, or even make a difference atm. But may be worth a shot?
Also: you can try the
EQEmu Launcher, it is great for keeping track of downloaded files needed as well as starting up the game.