It isn't work for good reason, you are trying to force a digital signal (HDMI) through an analogue connection (DVI-VGA adapter). The wy you have it hooked up will only work if there is something in the chain to convert the signal from digital to analogue.
If you're going to use VGA you need to find out which of the two DVI ports on the video card supports VGA, put the adapter there and use a VGA cable between the two. Or you could just use and HDMI cable if there is a spare HDMI port on the TV.