Switch – How Do I Fix This Broken Switch

Since there doesn't appear to be any grounding conductors in the box, and the box appears to be plastic. You've done all you can do, aside from installing a nonmetallic cover plate.

This answer might also be helpful.

If you're concerned about it, consider that it's been operating without a ground since the switch was originally installed. If you're really concerned, you could install a GFCI breaker to protect the circuit. If you're really really concerned about it, you could install an equipment grounding conductor from the panel to the switch.

Single Pole Switch Diagram

1. Turn power off at electrical panel by turning off the breaker.
2. Connect the two black wires as shown.
3. Connect the ground bare wire to the green screw making a clockwise hook so the ground is nice and secure. And tighten.
4. When you push the switch in make sure the ground is tuck back out of the way as to avoid any problems from ground faults.
5. Make sure the switch is in the correct upright position.