You mean one of these?
To be honest, I don't know what that part is called. But they do probably sell it on mouser. You can buy it or reuse what you have but then it is still a ton of work to fix without many tools. You can poke out those wires from the gray thing with a needle and crimp the wires then plug them back into that gray thing.
What I would do for that is convert it to a micro/mini-b by soldering some wires onto something similar to these and glue that mini/micro usb to the side of the reciever. Doing that will allow you to plug it into the machine using a usb cable or just using a usb adapter and have it right next to your Surface. If you want to use this adapter you will have to rip or cut the shield off the female side so it can be plugged into your Surface.