Are you using a National Instruments DMM (NI 4060 or NI 4070)? Are you trying to monitor the voltage across the LEDs? You're going to need to explain your setup in more detail.
I would suggest to look at the current to best detect whether an LED is on/off. Most LEDs have a specification of the current it takes to turn them on. You could monitor the current passing through the LED with a DMM and if it gets to the specification value, you will know the LED is on. I hope this helps.
Regards,
Todd D.