So with your internet connection being slow, i can understand not wanting to wait several days to redownload a game. This isn't guaranteed to work, since in some cases things like updates can cause problems.
Note: if it has downloaded the files already, you need to think carefully before doing these steps or you may have to start the download over again.
- Backup the old game files somewhere
- Delete the game via the steam client
- Check the game folder and delete the rest of the files/folders
- Copy the backed up game data files/folders over to the game folder again (close steam client just to make sure)
- If you see a InstallScript.vdf file in the game folder, delete it.
- Have steam install the game.
Best case scenario it sees the game files and says it's updated. Worst case, it still rejects the files for whatever reason and wants to redownload it. You may also have luck just having it validate the game files and it decides to work, but I did see one person say that if you have it install the games files in steam, pause it when it starts to download the game, copy the game files over to the game folder, and then validate the files it may work.
There doesn't seem to be a way that works every time, but if you need to validate the files, right click on the game in steam, go to properties, go to the Local Files tab, and then click Verify Integrity of Game Cache.
The only other thing I would suggest is just to let it download the game and find something else to do until it finishes. It's tough, but sometimes it's better to be patient than to waste your time.