What model Janome do you have? I have the 6600 and have no problem with the King Tut threads, or even Aurifil 28 wt!! I have also had no problem getting a heavier Connecting Threads Essential through the tension discs (but stopped using that thread because of all the fuzz it creates down in the needle and bobbin area!).
I suspect your discs might need to be cleaned. Perhaps you were using a thread with a lot of fluff, and the fluff is now embedded in the discs??
King Tut is a great thread, and IMO not too "thick" to get through the tension discs.