That laminate was originally glued with contact adhesive. You can buy a very small bottle of contact adhesive almost anywhere and it usually comes with a brush built into the cap.
Coat both surfaces with the adhesive then slide a penny in there on its edge for about twenty minutes to allow the adhesive to dry (flash). Once dried remove the penny and the two pieces will bond together again.
Here's a little trick you might try first and it won't cost you anything. You might use a hot iron (clothes iron) to try to simply press the piece back onto the door. The heat may just activate the old glue enough to fix it.