Carefully locating the hole with the marking of the missing piece, drill a hole suitable for a screw of about 3 mm diameter. The screw needs to extend about 10 mm from the surface of the lamp cover to actuate the switch, but it should not extend much beyond this.

The screw goes through the cover, from the outside of the lamp cover and must extend into the slot mentioned in step 1.

All better! The projector now works as intended.