In her book, Retrospect of Western Travel, Volume 2, Harriet Martineau, a writer and sociologist, collected the accounts of those who were, or claimed to have been, there when it all occurred. Martineau writes that the story began when Delphine LaLaurie was observed by a neighbor chasing a young slave girl with a whip and eventually driving the girl off the roof of the manor home, killing her. The neighbor reported this to the police, as mistreatment of slaves was a crime in New Orleans at the time. To elaborate more and answer your question about that, people at the time in Louisiana had something called the "Code Noir" which was set out by the French monarchy and laid out how they could/should be treated and gave them some small amount of legal protection. Article 42 says,

"Only the masters can, when they believe that their slaves will have deserved them to chain them and make them beat with rods or ropes. We forbid them to give them torture, nor to make them any mutilation of members, under penalty of confiscation of the slaves and to be proceeded against the masters extraordinarily."

And article 43 says,

"Enjoin our officers to criminally prosecute the masters or commanders who have killed a slave under their power or under their direction and to punish the murder according to the atrocity of the circumstances; and, in the event of absolution, allow our officers to dismiss both the masters and the absolved commanders, without their having to obtain letters of grace from us."

So, the complaint of cruelty was made under those statutes and the police investigated and found that she kept her slaves in deplorable conditions. They then removed the slaves and resold them, though LaLaurie had members of her family straw-buy them and quickly had them returned to her in secret, where she continued to torture them. LaLaurie kept her cook chained to the stove. The cook, obviously not wanting to live such a life anymore, attempted to commit suicide by burning the place down around her. When her neighbors rushed to help, some of them seized the opportunity to investigate the rumors about her and found out what she'd done with the slaves that had been returned to her. According to the accounts collected by Martineau,

"They were chained and tied in constained postures; some on their knees, some with hands above their heads. They had iron collars with spikes which kept their heads in one position. The cowhide, stiff with blood, hung against the wall; and there was a step-ladder on which this fiend stood while flogging her victims, in order to lay on the lashes with more effect. Every morning, it was her first employment after breakfast to lock herself in with her captives and flog them till her strength failed."

Slavery was a horrific institution built on cruelty, but Delphine LaLaurie was just so unfathomably evil in nearly every way that she transcended even the "everyday" cruelty of a society which systematically destroyed an entire race of people. She really was just that bad and there was just no way to rationalize it. You can look anywhere throughout accounts of the time period and find people making "manifest destiny" arguments about the inability of blacks to govern themselves and the right that gave whites to literally own them, but there wasn't even the thinnest veneer of that in this case. It was just completely unspeakable cruelty in every single way. It was so bad because it just couldn't be justified. It was just cruelty for the sake of it. Slavery and slave owners were cruel, but Delphine LaLaurie was really in a league of her own. She even abandoned her own daughters who were forced into poverty by the mob's destruction of their estate.

Whew, that's my first ever answer here. I hope it was up the standards of the sub! I tried my best!