This is a notorious place for people to get stuck in this game. Yes, you must have neglected to do something, and it's quite possible to work yourself into a dead end.
Don't worry about the slider puzzle.
What happens in SCK is that the sequence of conversations is important, and you could have used a conversation thread "too early", thus locking out other threads, one of which was essential to triggering the appearance of the letter at the Diner.
I suggest you back up before the teacher's loungd and the boiler room, (hope you have a save)and replay from there, using the walkthrough as a checklist. If you look in my walkthrough, find the picture of the dials in the boiler room, and carefully follow the conversations starting with Hal and the essay. Watch for the underlined clues in the WT. They are the essential sequence.
Hope this helps.