First off - You weren't kidding when you wrote "exploded". Components don't do that for no reason, so even if you find a replacement, you still have to make sure the rest of the board is well enough for that to not immediately kill itself too. It might've simply been hit and physically shattered, but best to make sure.

You didn't include a wide angle shot of the entire board, but from what I found online the missing part is close to one of the two DVI ports, and there seems to be a second triplet of identical components/circuitry for the other port. If your entire issue is caused by just this one missing part then I'd suspect it's limited to that port, and would move around when you swap in an intact part from the other set. Otherwise there might be even more wrong with the board. Does the other port work?
Also, did you try an analog cable instead of digital (if the card supports that - haven't checked), or vice versa? If just one of the signal types is affected, that'll help narrow down what the broken part did.

That's a common mode filter. Before bothering to find a replacement, just solder two wires like this, and check if the rest of the card actually works:

It's more likely that the filter is on the VGA lines than the DVI lanes.