Azulas story did not end with her being defeated by Zuko and Katara in the Agni Kai. After being named Fire Lord by her father, Ozai, her paranoia and narcissism evolved into a full-blown mental breakdown. When Zuko and Katara arrived to challenge Azula's claim to the Fire Nation throne, they found her deep in a rage. Zuko used her instability in his favor during their Agni Kai, and with Katara's help, managed to imprison his sister and take the title of Fire Lord.
Azula's fate after the events of Avatar: The Last Airbender is explored in the comics The Search and Smoke and Shadows. In Smoke and Shadows, Azula had been secretly plotting with others to overthrow Zuko as the Fire Lord. However, Zuko's reign was still fresh, and many in the Fire Nation were upset with his appeasement policy. Azula saw a golden opportunity to change the leadership from the inside and gathered loyalists to her father.
In conclusion, Azula's story did not end with her defeat in the Agni Kai. She suffered a mental breakdown and was imprisoned by her brother, Zuko. Her fate after the events of Avatar: The Last Airbender is explored in the comics The Search and Smoke and Shadows.
