I just wanted to add some more things that I have tried. I have pressed Windows key + P and chose Extend, but that did not work. I unplugged the monitor that is on and the monitor that is currently off is now on (so it appears both HDMI ports are working and both HDMI cables are working). When I plugged the other monitor back in, only one works. I have once again completely removed the Catalyst software and the Graphics Driver, rebooted, the reinstalled the latest driver (15.200.1062.1004) but still did not work. I should also mention that when I open the Device Manager, only one of the monitors is showing under "Monitors".
I am going to keep trying and hope something works. I need my second monitor. =)
Okay - so I got it to work. Kind of. Here's what I did. Disconnected both HDMI cables, connected the DVI cable to monitor 1 and a HDMI cable to monitor 2. At first nothing happened. I went to control panel, typed in Troubleshooting in the search bar, clicked on Troubleshooting on the top of the newly populated screen, chose Hardware and Sound from the new screen, then clicked on Hardware and Devices. Almost immediately my second monitor came on and now I am enjoying two monitors again.
I have not tried going back to using two HDMI cables yet. I will do that later tonight and hope that it works. But for right now, it seemed to have fixed my problem. I hope this post in some way helps someone that may have this issue in the future.