First before anything be clear to everyone that you are serious about changing the way the server room operates, and take in everyone's input about how to make it better. You might find that the staff using it have a good idea. If a "strong person" can get in, maybe it is as simple as fixing the lock.
I would recommend implementing a security fob system to accomplish this. Then each user will have a fob assigned to them that unlocks the door. This will track exactly who was in or out. Unfortunately this sounds like it is a more costly solution than you are looking for.
Keyscan is popular for a low end system and somewhat low cost/maintenance. http://www.keyscan.ca/English/Security_E.html
Consider implementing a webcam pointed at the door snapping pictures every few minutes, or a more advanced one that can do motion tracking.
Another way to do it is with light detection. When leaving the server room turn the lights off. You can use a light sensor such as http://www.eesensors.com/Websensor.html to monitor the light values, and if integrated into a nagios monitoring solution can email you whenever someone enters the room. Use these emails in conjunction with the sign in sheet to ensure people are following the rules.
With all of the above as you say implementing a sign in/sign out system would be a good idea.