GU10 socket re-wiring
If you were to go to an online retailer such as Amazon looking for a replacement GU10 Porcelain socket you will quickly learn a few points:


They are not particularly expensive. Often batches of then are showing for 8 to 12 USD. Individual sockets for in the 3 to 4 USD range.
Virtually all of them come with two short pigtail wires that are crimp connected to the socket. (Unless the unit is a socket type adapter to regular Edison bulb base).
There are two general styles. One is the plain porcelain socket and the other has a metal bracket on the rear to attach it to a threaded tube type mount.


For your replacement get the appropriate type of unit with its pigtail leads. Then:


Demount the light fixture shroud from the ceiling and disconnect the old lamp wires from the permanent wiring in the electrical junction box. (These old wires would be connected with wire nuts or other type of wire coupling depending upon your region).
Clean up the fixture shroud and mount the new socket to the shroud with the pigtail wires sticking out the back.
Connect the new pigtail wires to the permanent wiring in the electrical box. Use new wiring couplers or wire nuts if needed.
Remount the light fixture shroud up to the ceiling.