I had a similar problem. I resolve it by reinstalling the monitor using the drivers on the LG disc instead of plug and play drivers - I also installed the Forte Manager that came with it. I am using a DVI cable rather than a VGA. Have you tried a different cable.
As the new PC can connect via HDMI, is it possible that there is a setting to select which output mode the graphics card uses? Obviously this is problematical as you'd have to connect to the TV via HDMI and then change the setting at which point you might lose the existing display, unless you can select multiple outputs.