i use an MX-980 in my HT setup that has my 360 in it, but can confirm the 360 power command is a little sketchy. i often have to press power twice. my work around was to simply make a 360 power toggle on my main activity page so i can power on the 360 before even changing activites (this also allows the 360 to load up to the main screen before i switch inputs). i dont believe harmony's programming would allow for this work around, so i would suggest just having the xbox power toggle on the 1st page of your 360 activity pages.

all the best!

edit:sorry OP, are you saying the 360 will not respond AT ALL? if so then there may be a glitch in your software or unit itself. it might be worth a call to logitech, as they do have an excellent warranty process if your unit needs to be replaced.