The solution to this is fairly simple. I'll guide you through the process of taking the back housing of the HTC One X carefully without breaking anything.
What you need to do first is get the replacement power button ready. You can get these on eBay. You are probably better off buying the whole back cover black plastic housing altogether with the power button and volume button. Here's a link to it:
http://www.ebay.com.au/itm/New-Full-Housing-Battery-Back-Case-Cover-Black-For-HTC-One-X-S720e-G23-/351258759126?hash=item51c8a74fd6:g:81gAAOSw2XFUlAJ9
Make sure you double check the model no if it is correct / corresponds to your one.
If you do decide to get the part and replace it, you can follow this guide to step 2:
[guide|31412]