My guess is that it's an illuminated cover, which no longer works. Turn power off, disconnect the two wires to the cover, and replace with a regular cover. While you're at it, you might as well replace the switch.
The holes in the plaster ears of the box will take screws very well. That is what usually holds the box in place when it is mounted in a cabinet.
To allow the cover plate to set well, you could use 4 gauge screws or a larger diameter and countersink the holes in the mounting ears a bit to help let the receptacle set tight to the face of the box. You would do well to break off the ears on the receptacle at the score lines to help clear the screw heads too.
Best Answer
Those ears snap off
Those are called "plaster ears" and can be broken off using a pair of pliers. Snapping them off will let you fit the new switch into the existing box.