While dusting for prints can be a time sensitive action, you should be able to analyze prints any time of day or night. While I've never had this exact issue, I have had issues similar. So I'm not entirely sure they'll work, there's a couple things you could try:
It could be a simple routing issue. Try going to the sidewalk in front of the police station and then selecting "Analyze Prints."
You could try resetSim. Sometimes when a sim is acting goofy, this can fix it.
Or perhaps the rabbit hole itself is bugged. Try selecting "Do Low Level Police Work" and see what happens. Your sim will either make his/her way to the police station, or the action icon will disappear like it does when you dust for prints. If the latter happens, the police station itself is likely the problem. Try deleting it via edit town, and placing a new copy of it in the same spot. I've personally never had to do this, but I've heard tell of people having similar issues with rabbit hole buildings. Either way, it might not be a bad idea to save before trying it.
If none of these fixes the problem, the case is probably bugged. This has happened to me numerous times, and it seems to be a regular issue with the investigator profession. So unless someone else has an idea, it would probably be best to just cancel the case and get another.