First let me summarize the thread. You are trying to connect your computer to your TV via an HDMI port, with no luck with HDMI to HDMI. Your graphics card is likely an ATI Raedon HD 5770.
Alright - You are in luck, since I have done this a few times before. First off, go into your TV's menu. If there is an option to disable/enable ports, check those and enable any disabled ports you are trying to use (or just enable all of them for testing purposes).
When I connected my Gateway FX 6840-03e to a TV, the system automatically detected the the TV as a monitor. I confess that I didn't have a regular monitor hooked up, but it worked.
Anyway, if your computer doesn't auto-detect the monitor, go into Control Panel -> Display -> Screen Resolution and click Detect. It should pop up on the window. If not, then you may need a new cable. If a second cable doesn't work, try another port.
If all else fails, your TV is probably messed up.