Hi Chumdo,
I have a Kenmore two door ice/water fridge that’s been renamed as a Whirlpool and I had to remove the front cover plate so I’m thinking there’s a chance your cover is fitted in a similar way as mine.
If that’s the case you have to be very careful well removing the cover plate so you don’ break off the small plastic snaps that’s part of the mold of the cover plate. They are small bits of plastic that stick out at an angle and slip onto grooves on the fridge and they hold the cover on the door.
The safest way to remove the cover is to look underneath the cover and at the bottom, and you should see two small slits just big enough to slip a knife into, with the knife gently prior the cover out and down so you avoid breaking the plastic snaps. Once you have released the bottom of the cover carefully lift the cover up so you don’t break to top plastic snaps.
My front cover had two snaps at the bottom and three at the top which I accidentally broke off, so I used doubled sided masking tape at the top to hold the top of the cover back on the fridge.
When the cover is off it should be fairly easy to work out why the light is stuck on. The plastic slider that sides back and forth to switch the light off/on may have broken or slipped out of place.
Hope this can help you out. Let us know how you go
Regards from Noelene