Thermocouple connection question
Two things.  First, a thermocouple is not the best way to measure LED temperature.  Thermocouple signals are very small.  The main advantage to thermocouples is that they are simply two metals, and as such can stand high temperatures.  Thermistors give you much bigger signals and can still cover the temperature range of a LED.

Second, LEDs are diodes and therefore have a significant temperature dependence themselves.  If the LED is driven with a known current, its forward voltage drop will tell you its temperature after some calibration.  You said the LED is driven with PWM.  What does the driving circuit look like?  Is the current reasonably fixed when on?  If so, you only need to measure the voltage during the on phase to determine temperature.  You will have to test this at a few known temperatures to calibrate, but once that's done it should be pretty repeatable.