Right. The HDMi port is on the actual motherboard.
The graphics card which your VGA cable leaves will prevent the motherboards own graphics from working, hence there's no output from your HDMi port. That's how it works, you can have one or the other - but not both functioning together.
You have two choices.
1) Take out the graphics card, load the drivers for the motherboards own graphics, and connect both HDMi and the VGA/DViD to your TV and monitor respectively.
2) Fit a better graphics card which has HDMi and VGA (and probably DViD) outputs, load the drivers for it, and use that instead.
Kudos to Duzzy for the brainwave...