Joffrey probably sent the assassin, inspired by Robert drunkenly claiming it would be better to pt Bran out of his misery. Though it was never proven or even guessed at until long after everyone who cared was dead, Tyrion & Jaime each independently figured it out, and Tyrion halfway confirmed it; after Joffrey made a comment about being "no stranger to Valyrian steel" he described the dagger and Joffrey recognized the description. Jaime, annoyed that Catelyn believed he or Tyrion might have done it questioned Cersei, and her description of the Robert comment made him realize it was Joffrey. That's also his assumption of Tyrion's motive for killing Joffrey, as revenge for getting the Imp kidnapped by Catelyn.

Of course, all of that is in the books. IDK what the show is doing.

I apologize for validating Cannoli, I know that is the one thing we must never do! (besides feeding him after midnight )