I might be wrong here (this is just as I understand it

), but I think that the DVI socket on a PC GFX card not only outputs the single stream DVI (digital) signal but the RGBHV required for VGA (analogue). Thus, when adding the adapter (DVI - VGA) it is simply selecting the pins that are outputting the analogue VGA signal so you can use a VGA type monitor.
So if you were to get a VGA - DVI adapter I doubt that there will be a digital signal output stream (DVI) that can then be adapted to the HDMI socket format. The GFX output will just be an analogue RGBHV signal.
The best option would be to get a cheap(ish) DVI GFX card for your PC. I doubt it would cost much.
Have you not got a VGA on the back of the LCD anyway? I use mine XBOX360 > LCD via VGA... looks great
