OK, you'll need to get the multimeter out and check if it is AC or DC 12 V as per Visitor's comment.
If it is DC, as a test, you can wire a resistor in parallel with the LED cob to increase the current draw - start with a 120 Ohm 5 watt resistor - it should draw an extra 100mA in addition to the LED current.
If the LED stays on, then you have the solution, although some of the advantage of moving to an LED is lost because of the increased current.
On the other hand, if the supply is AC, I would expect the LED to run at ~half brightness, rather than come up bright and then go out. The fix for that is to add a bridge rectifier to the supply to convert it to DC.
A picture or two of the actual desk lamp would help too.