How do I mount this non-standard wall plate to a completed drywall wall?
You can use a low-voltage retrofit mounting bracket, which is the same size and has the same screw dimensions as a standard outlet box, but without the "box" part. 



Because it's low-voltage, you don't need the mechanical protection of the box like you do for high-voltage. It also gives you a lot more room to work with, which is very nice for bulky connectors like the HDMI and VGA ones you have.

After you cut the appropriate sized hole in the wall, you mount the bracket and it clamps down onto the drywall. Then you can run the wires into it and attach the faceplate, just like you would with any box mounted in the wall.

Here's what it looks like in the wall (this one is orange, but otherwise the same thing): 





Note, there are also low-voltage mounting plates for new construction, which are basically the same thing but go up before drywall and attach directly to a stud.