Hi Kyle,
It seems that you've almost done everything to resolve the issue with your Surface Pro 3 not able to recognize the Type Cover.
Also, since you've already tried a known good Type Cover and gives you the same result, then we can say that this is a possible hardware issue. Performing a factory reset would be the best way verify this. However, this will wipe out all of your files.
In case you decide to do a factory reset, make sure that you back up all your data. Click on this link and refer to the Reset your Surface section.
If you were able to verify that the issue is a hardware failure, you have the option to get your Surface device serviced. For more information on how to get service for Surface, click on this link.
Feel free to post back if you have any concerns.