You cannot tell me that Iroh would not show the exact same patience and compassion for Azula that he did for Zuko.
After the show was over, there is no doubt in my mind that he visited her at the asylum once a week, bringing a cup of his best tea and a Pai Sho board.
The first few times, she threw the tea in his face. He kept coming back.
Eventually, she would take the tea and insult his Pai Sho technique. It’s several dozen more visits before she will attempt to play.
Iroh knows that Azula, as the “favored child” of a monster, is as much of a victim of his brother’s twisted parenting as Zuko is, and he won’t rest until he’s helped her heal as much as possible.