Just to clarify, is your phone not turning on at all? Or is it an intermittent issue?
To test whether it's your power button or another issue, remove your charged up battery and insert it in. If it is the power button that's the issue, (and not the board), then the phone will turn on anyway. If it doesn't turn on, then you may have another issue.
We don't have a HTC Desire X repair guide yet, and I'm guessing the power and volume are real buttons, not touch capacitive. May have to wait for that, or you may have to teardown yourself and replace it with the part. I've found the power button here.
http://www.aliexpress.com/item/For-HTC-Desire-X-T328-Power-Button-Switch-On-Off-and-SIM-Card-Slot-with-Flex/32489211723.html