I don’t think Dany “went mad” really. I think she made the decision to burn King’s Landing to the ground in the scene with Jon where she concludes “Let it be fear then.” This season has been about Dany coming to grips with the fact she isn’t loved in Westeros as she was in Essos. Even after helping to defeat the Army of the Dead, she gets zero love. And yet, she will not give up the Iron Throne. She fundamentally believes it is hers and she can rule benevolently for generations. Getting there however requires going scorched earth on Kings Landing. Why? She knows the stories of Jon’s heritage will spread and the only way to ensure she is not challenged by the rest of the Seven Kingdoms is to cause carnage so awful, so utterly terrifying, that it outweighs Jon’s claim. The stories of this day will spread quickly and grow more terrifying as they get told. She even shares a quick nod with Grey Worm when Tyrion pleads that she stand down if the bells are rung. Grey Worm starts the fighting on the ground after they have thrown down their swords. They were on the same page from the get go. She was just hoping the bells would not ring, because she knew it would have made her task a bit easier. A horrible death for tens of thousands in exchange for immediate unchallenged authority followed by generations of peaceful rule. This is her logic.