-( Story related, cannot be missed )-
Kanji's dungeon, the Steamy Bathhouse, becomes available starting 05/19. You have until 06/04 to clear the dungeon and rescue him, or else you will be experiencing a Game Over early on. However, this starts your investigation into key factors behind the missing, as before you simply entered the Midnight Channel and found access to the dungeon waiting for you.
1) To unlock the dungeon, you will need to acquire intimate knowledge behind Kanji. His mother would be the best to find this out, and from there, the other key witnesses will be a lady in the lobby at Junes and, after that on specific days, the youth Kanji was seen talking to before. With the key information obtained, return to Teddie to unlock the dungeon.
2) There are 11 floors to this dungeon. It is fairly straightforward with taunts from the Shadow leading you through.
I'll leave the boss to you to figure out. He'll be with two adds, Nice Guy and Tough Guy. If you want to take care of the adds before focusing on the big boss, take care of Nice Guy first, as he has Red Wall (making Tough Guy resistant to fire rather than weak) and Heat Riser (making whomever he uses it on hit harder, more resistant, and more evasive).