I did the same, using a 1W Stanley LED lamp. Used an old LGA775 heatsink without the fan as the heatsink, tapped 2 M3 holes for the screws. Runs from a slightly modified Nokia charger, with a 4R7 resistor as current limiting. Had to replace the supply caps on the charger ( bulging and near to bang) primary and secondary, with salvaged old but good low ESR capacitors. Added a steel block in the bottom to make it stable, and it works well. Not as bright as the original 12V 50W GU5.3 halogen, but as bright as a 20W halogen. Just a lot cooler, now it is a lamp and not a heater, and the plastic parts no longer deform from heat. The heatsink is a little overlarge, as the die is only 5C or so above ambient. Now a nice cool running Anglepoise style desk lamp.
I am looking to do the same to another lamp, with the exposed wiring bars like the OP, using some LED retrofit lamps that are supposed to replace a halogen. That lamp already has a replacement toroidal transformer in it, the old one went crispy from overheating when somebody put a 50W lamp in it instead of a 35W lamp. Old core made a very nice current transformer core though.