This one can be tricky if you don't realize it. After you fight Mr. Tick Tock, you proceed into a very dark room, and are quite immediately ambushed by 3 Bronto Burts. Now, what you probably don't see is that there's an unmarked door right next to the slightly raised ground just before those flying foes. While you can enter it blind, to see it you would normally have to use the Light copy power. You can grab this from a Cool Spook, which looks like a glowing ghost with sunglasses and a halo (it may look like a book at first, though, haha), that is found farther into this dark room. See it or not, enter the room to find your secret switch!
Using Light will light up the entire room, which often will reveal a door that may have been hidden by the darkness. So, going forwards, if you find yourself in a dark room, seek out a Cool Spook and make sure you don't miss anything.
The setup actually attempts to dissuade you from investigating there in this particular room. By presenting you with Mike in the previous room, and then assaulting you with multiple foes, it entices a player to use the ability and charge forward without searching. Devious!