Why does an LED in a Can Light go out after Several Minutes?
Something with a time delay like that is usually thermal.

The heat of the halogens is killing the LEDs

Halogens want to burn very hot. They are deliriously happy inside ovens. They wouldn't even mind lighting up a steel mill ladle.   I'm told junction temp on a halogen is 2000C. 

LEDs do not like heat. You own a screw-in light bulb product based on LED technology.  It has an electronic driver and heat sinks.  Deep inside it is LED emitters proper, they are much smaller than a fingernail.  They need to stay under 85C, even though they're making about 5.5W of heat.  That means there needs to be heatsinking/heatpiping to carry heat away.  Thermal movement down a heat sink is proportional to the difference in temperature.  So that heat sink must be significantly cooler than 85C or else the heat won't move away from the LED emitters fast enough, and they will be destroyed. 

You have one LED screw-in product in an enclosure or ceiling space containing a bunch of halogens.  The ceiling space might not get up to 85C, but it doesn't have to; the heat sink will be defeated well before that.  I imagine these products have a temp sensor on the LED or heatsink, and are shutting down to prevent the LED from incinerating itself.  

The way to test this for sure is unscrew all the halogens and run the LEDs alone.  If it stays stable, that was it.