On this page of the Disco Elysium guide, you will find a detailed walkthrough for the Explore the Whirling's secret passages quest.
Your goal in the Explore the Whirling's secret passages quest in Disco Elysium is to open and then examine the mysterious room at the back of the cafe in the Whirling-In-Rags Hotel. You can get inside in two ways.
Starting the quest
You activate this quest when you enter the kitchen and interact with the door visible in the image above.
Entering through the kitchen
When the second day comes, go to the cafeteria and confront Titus and his men (Hardie Boys) - Interrogate the vigilantes and Prove your authority to Titus Hardie quests. Examine the window marked in the image above - if you successfully pass a very difficult Perception skill check, you will notice the key outside that can open the locked kitchen door.
Entering through the terrace
After entering Klaasje's room, you can go to the rooftop terrace where you will find a locked door. They can be opened by force but this requires passing a very difficult Physical Instrument skill check.
Exploring the secret room
Old broken pinball machines are stored On the ground floor of the secret room. By interacting with one of these devices, you will learn interesting things about Kim.
Upstairs you will wind find some sort of office. Pay attention to the footprints someone has recently left here - when you interact with them, Harry and Kim will conclude that it may have to do with the case. The quest will be complete.