Once the "Donut Ring" scratch has been burnt into the disc, it can very frustrating to attempt to get the game working as you every time you attempt to, it gets further and further into the game before rejecting the disc and saying "Disc Unreadable", however it is a very low possibility you can recover disc data from that disc.
Is there a way? - Of course!
Attempting DIY is an option and worth trying because at the end of the day, the disc is not going to work by itself, but in your circumstances I can see the problem. My suggested options, although they may not apply in your location:
In my local area with GAME, if you purchase a game you have up to 14 days to return it as long as the product is in the same quality and standard as when you purchased it. Therefore I have done this in the past and although it is frowned upon, has no real harm. Purchase a pre-owned FIFA 14 from a store and download it, after the download is done try your FIFA 14 and if it works Hoorah! Just return it with the excuse that your console can't read the disc.
If that isn't an option for you my only other suggestion in your case is to buy the game again, annoying but we have all had to deal with it. Personally I have had to experience this around 14 times, every single time the same reason; I charge my controller using a play and charge kit, wire wraps around my foot and as I stand up my Xbox console launches off the desk. However reality is reality and we all have to face it, some of us are more lucky than others. However, if you don't feel like buying another FIFA 14 due to price, purchase another game for a lower price to keep you entertained until the price has dropped.
Really hope this rambling has helped, just a few suggestions for you. Really hope it all works out for you!