I think (not a mental health professional) that AW is absolutely damaged and has mental health issues. I also think he is not legally insane.
His abducting Lizzie from the home, his hiding of her bloodied clothes, his hiding her body, his lies about whereabouts to LE, his attempts (uggh) to lick the dried blood evidence from his hands while sitting in the interview room, are all IMO evidence of his consciousness of guilt both during and after the crime. ie. He knew what he did was wrong and therefore took pains to hide it and try to distance himself from it.
I think he is an angry young man. I think he may have been intensely jealous of Lizzie's childhood, his sister's little family, where from all reports there was love. I think little Lizzie represented to him all he had been denied as a child. She was loved, she was protected. Nobody ever did that for him. Why should she get that?
My heart breaks for Lizzie's mama.