Question: how did baelish kill joffrey?
Answer: In reality, it was Olenna who had poisoned the wine. She had arranged with Littlefinger to end Joffrey's life and used Sansa Stark in the process. Littlefinger had worked with Dontos Hollard to deliver the poison, which was hidden in a stone attached to a necklace given to Tyrion's wife by Olenna.