I really don't understand the problem here. She literally had no way of knowing for sure if he died. It wasn't just Zuko's attitude that tipped her off, but specifically his answer to her question. He spent more time hunting the Avatar than anyone, out of everyone he would know if there was a way he could've survived. She knows when Zuko is lying. It's a win-win situation for her to tell her dad Zuko did it.
It wasn't an attempt to redeem her. It was just so we can understand her character, why she acts the way she does. She didn't suddenly become a good person, like you are suggesting how she should've been kinder to Zuko...even if that wouldn't make sense considering her attitude to Zuko up to know. Aside from literally trying to get him killed, she also taunted him when she learned Azulon was planning on killing Zuko while they were kids.
She was spared because she was chained up. Why should they execute her? Zuko made no attempt to reason with her, he came there to waste her ass and he would have if Azula didn't bring Katara into it.
People will read this argument and come to their own conclusions.