Sounds like you haven't received the full hypnosis equipment(the ring you get from Shady later, the lantern is stored with Katara). I've just caught up with the house burning down so it shouldn't be long to get you passed that delay.
Go to the Arena to try and speak with Shady, but meet his girl instead>Go speak with June>Go speak with Iroh>Go speak with Suki>Go speak with Joo Dee>Back to Suki>Speak with Toph(and assist with her night terrors)>Train with Toph>Chat at night(should be 9/16)>Go through the tunnels(straight to end through busted wall, right, up, left and into door), free(or not) the three girls huddled up inside>go train with Toph, then speak with her at night>Next day go speak with Iroh about his book>Speak with June and pay her for Iroh's book>train until night, chat with Toph(pick the pro-Toph/relationship answers)>Train, then speak with Toph and go to sleep. If you've done everything right to that point, you should have the option to rebuild your house the next day when you click on it.
After that it'll trigger a cutscene. Choose to help Suki and upgrade the tavern. Then speak with Suki inside, and she'll talk about the lantern. After that, you should be all setup so that the next time you enter the tavern, it will trigger the Dai Lee scene that will start Suki's de-hypnotizing process.
