Passive switch that detects when a door is opened
Then a friend suggested that I put in a button that would be pressed when the door was closed, opening the circuit with the LEDs, and closing the circuit when the door was opened. This would be very battery effective, but it is hard to place a button correctly, and it would propably come off if I accidentally closed the door with to much force.


I think you really should use a normally closed push button as your friend suggest and tackle the drawbacks mechanically:  

Use e.g. a guided rod in combination with a spring to press the button.
The spring can solve the mounting misalignment issue to some extend as well as the slam-the-door issue.
EDIT: Of course those have been invented already: search internet on "spring rod limit switch".


Source (for as long as it exists there): www.alibaba.com

Another option is using a roller lever switch like shown below:  



Source: Technical Guide of Basic Switches by Omron