In the Library section, navigate to the game and click on the three. If steam sees it, then you can verify cache and steam will correct any errors in the. Steam will verify the games files - this process may take several minutes. Change id to 345350 (that's the id for lighting returns), name to 'LIGHTNING RETURNS™: FINAL FANTASY® XIII' and install directory to whatever you called the folder and bytes downloaded to something other than 0. What refreshing Steam files actually do is delete all the configuration files of the Steam client and then force it to install them again.
Furthermore, your user data will also be preserved. I would try putting the files in steamapp/common/gamename, then use this as a template for your. When we refresh your Steam files, we will preserve your downloaded games so you won’t have to download them again. I don't think steam makes one until the game installs, so you will probably have to make your own. acf for them as well, in the steamapps folder. So, if a game has 3 files, game.exe, config.ini and background.jpg, including a file called game01.sav would be ignored by the verification process. If you already have the game files, then they are supposed to go in 'steamapps/common/gamename' and have a releavant. Verifying a game's files, whether through Steam (for Steam games) or Galaxy (for GOG games) checks only the files on the game's manifest.
Downloading folder is where incomplete downloads temporarily go, you can't add stuff and make steam see it.
The game cache is where the game is installed, that's why it won't see them. Steam won't verify it, because the 'downloading' folder is not the game cache.
