Where has the socket failed? It still looks usable to me.
If it was my fitting I would replace what appears to be either a 10W or 20W halogen lamp with a 2 or 3 watt LED equivalent available from your favourite hardware store etc. Future overheating issues eliminated.
If the socket has failed completely I would cut the wires just beyond any hardened sheathing and connect the wiring directly to the LED pins with suitably sized small connector blocks.
There has been heat there by the look of the pins on the lamp so I wouldn't risk a new lamp without redoing the the termination, one way is as Scubadoo says or you can get replacement sockets and leads from any electrical wholesaler!