Aside from previously mentioned missing or damaged components near DVI area, you can also try to do this :
1) Connect GPU via VGA/D-Sub cable
2) Install drivers under Windows, and restart to check if full resolution is available on next boot
3) If previous step is successful, shut down PC and connect HDMI monitor (via passive adapter to DVI).
NOTE : Try to check if anything changes with HDMI to DVI-D one (unless you already have it) :
The attachment DVI-D vs DVI-I.jpeg is no longer available
^DVI-I adapters (right) are NOT compatible with DVI-D ports (left).
4) DO NOT disconnect VGA monitor at this point, just turn on the PC
5) Once in Windows - check if both monitors are detected, and if both can display image at native resolution
6) Shut down and disconnect VGA monitor
7) If everything goes OK, you should be able to see image from DVI port when Windows loads GPU driver.