Have you tried another HDMI cable or another device with the cable from you Xbox to make sure it is your Xbox?
At least yours is just the HDMI port. My CD ROM drive failed a year ago, and the motherboard failed last week.
The way the HDMI port is connected, there is no way just to “fix it” easily. I don’t mean to sound like a jerk, but by just asking that, you will not be able to fix it.
You have 2 options. First, if you have not broken the seal on your zbox , then you can send it in to be fixed “that is what I am going to do” it should cost around $100. The other option is just to buy a new xbox.
I know this is not the news you want, but it is the reality.