By carefully prying along a seam, it is possible to gently pry open the microswitch (i.e. it snaps together and is not glued shut).

The switch contacts (circled in red) might be dirty from grit or oxidation, causing the switch to fail.

You can clean the switch contacts by inserting a pice of paper between the contacts, then pressing the tiny plastic switch activator while dragging the piece of paper through the switch contacts (see second photo).

If the contacts were dirty, you might see evidence of this on the paper you used to clean them, as shown in the third photo.

After cleaning the contacts, use your multimeter to see if the switch now works correctly. If it doesn't, the switch may be broken beyond repair.

Before giving up, you might try using some fine grit sandpaper instead of plain paper, to clean the switch contacts.