Our house has motion-sensing light outside that come on at night when
motion is detected. The problem is that after a power failure, they
stay on all night (and maybe all day, I'm not sure). After a few days
of switching them off and back on,

Wait til dark. Turn OFF the switch..leave it OFF for about 20-30 sec.
Then turn it back 'on'..the light should be OFF (and re-set)