Two thoughts, from a mechanical engineer.

1. Cheap solution: write DOORBELL WORKS! on a 3x5 card and tape it to the door.

2. Also cheap solution: if you're like 90% of people you have a smoke alarm that doesn't work because you've changed the battery. Take out the little piezo speaker and put it on the other side of the door. Wire it to the buzzer. Include the 9 volt battery as needed. (not sure exactly how your doorbell is powered.

I apologize in advance for violating the Terms of Use of this forum, which include to provide the most complicated DIY electronic design project possible for any problem that might have a simpler solution.