Usually, Idol Manager's mods come in a folder. So, for example, if you download 'Mod A' and you want to put it in your 'mods' folder, you have to place the 'Mod A' folder inside the 'mods' folder, instead of exctracting 'Mod A' files from its folder and moving them to the game's 'mods' folder individually.
The weird thing, though, is that usually Steam Workshop mods are automatically installed and ready to use as soon as you subscribe to them, so technically you wouldn't need to download the mod's folder and place it in 'mods'. Would you mind sharing the name of the mod you're trying to use?
Although this mod is a bit different from the rest (as in it doesn't add/change textures, SFX or story content, but rather affects the game's code), it does show up here on my in-game Mods menu. All I had to do was subscribe to it on Steam Workshop and open the game. Does it show up on yours? Is it enabled?
Could it be that you subscribed to the mod, but didn't download it? That should happen automatically, though.
Also, according to the mod author, "The mod shouldn't have any compatibility issue if your other mods don't touch on the Business, Shows or Singles folders.". Could that be the issue?
No this is my first mod, I downloaded it, opened the game, and it didn't show up. So then I put it into the mods folder and it still didn't show up. It must have downloaded, cuz I found the file? I clicked on the in-game mods thing, waited for a bit (incase it was loading) and nothing showed up. Tried a couple of times.