Step 1: Make sure the game is completely deleted from steam. To be sure, delete local content of the game. Then you want to go ahead and start the download from steam like normal so it will get the appmanifest id and place it in the "steamapps" folder (this id is unique for each game, ex. appmanifest_8870.acf for Bioshock Infinite). Also, we want steam to start the download so it will populate the "downloading" folder in "steamapps" as well.

Step 2: Pause the download on steam and then go to steam-->steamapps-->downloading--> {insert name of folder}. The folder inside the "downloading" folder will vary depending on the game, but I believe it will always match the appmanifest id (ex. Bioshock Infinite appmanifest id was "8870" and the folder inside the "downloading" folder was "8870" as well). Once you have determined the correct folder, go ahead and copy/paste all your game files from your backup device.

Step 3: Once all files have been transferred, go ahead and resume download from steam.

Results: My download for both Bioshock Infinite and Borderlands 2 went from 0% to about 97%-99% almost immediately. I am not sure what files it is downloading afterwards, but it mostly recognizes the game after these steps. I imagine this will work for all other games that have this issue, but as of now I have only successfully tested this on the above two games. //

I had success with another process:

1: copy content from the backup into steam\steamapps\common*game name here*
2: copy appmanifest_###.acf to steam\steamapps\
Find number from google or other method
3: install game through steam, right after preallocating space it says 'Discovering local files' or something like that
4: download jumps to 99% or 100% (maybe less if you haven't played for a while and updates released.

Done