If you look at the Wikipedia article for DVI, especially the pinout, you'll see that DVI also can carry analog signals (the variant called "DVI-A"). That's why DVI to VGA adapters work: They take the the analog signals from the DVI-A pins, and route them to the corresponding VGA pins.
So if your graphics card doesn't support analog pins on the DVI connector (which it doesn't, it's a double link DVI port according to the specs, and in your picture it's missing the analog pins if you compare with the pinout on Wikipedia), then there's no way to convert it to VGA directly with a cable adapter.
There are other, more expensive adapters which are basically a small graphics cards which read the DVI/HDMI signal, and then produces a completely different VGA signal from that.
So use on of those, buy a different graphics card or use your old graphics card in addition to the new one, or buy a new monitor.
DVI and HDMI are basically the same digital signal. The main differences are that DVI has the optional analog pins, and HDMI provides for a copy-protection scheme (HDCP) that's necessary for some consumer electronics, but not an issue between computer and monitor. HDMI to DVI cable converters work fine both directions.