Why did Django kill Miss Lara?

Him hating all the white people is a bit of a stretch, after all that white Dr. Shultz did for him.

However, hating all the Candies is not a wee bit of a stretch. Covering his track is also a good motive: random slaves are not very likely reveal (or even be asked) who killed their hated masters when some sheriff or a marshal come to investigate, but Calvin Candie's sister surely would.

Then there is the matter of Django's punishment at the Candieland. It was her to decide to LeQuint Dickey Mining Company. This was the destiny waiting for Django, by Miss Lara's decision, as described by Stephen:

Lo and behold, out of nowhere, Miss Lara come up with the bright idea of giving your ass to the LeQuint Dickey Mining Company. And as a slave of the LeQuint Dickey Mining Company,henceforth, till the day you die, all day, every day, you will be swinging a sledgehammer, turning big rocks into little rocks.
Now, when you get there, they gonna take away your name, give you a number and a sledgehammer, and say, "Get to work!"
One word of sass, they cuts out your tongue. And they good at it too. You won't bleed out. Oh, they does that real good.
They gonna work ya.
All day, every day, till your back give out.
Then they're gonna hit you in the head with a hammer, throw your ass down the nigger hole.
And that will be the story of you, Django.

And it's not like miss Lara was really concerned about Broomhilda:

We are eating. Ain't no one wanna look at her whipped up back.

She's merely concerned about unpleasantness at the dinner table.

All taken into account, there is no reason for Django to be merciful to her, but there are quite a few to treat her like all the other white people at the Candieland.

She might not have ordered anything particularly severe that we saw but she was complicit with the brutal treatment of the slaves at the house, including Broomhilda.

Her brother adored her and she could easily have used that influence to ease the plight of the people there but she chose not to and merely enjoyed the luxury of her life off their backs.

She also, as Vendran pointed out, gave the order for Django to be sent to the mining company and was involved in the conversation about how best to torture him.

IMHO she was lucky he killed her with one shot and did not kneecap her first, she was as despicable as her brother.

He did it because she was the one who was going to have his genitals removed and then who decided to send him to the mining company as an even worse lifelong punishment than keeping him at the Candyland plantation. She basically was saying just killing him would be too easy, she wanted him to suffer for a very long time. She was Leonardo DiCaprio's sister too and complicit with all of the slavery.

Well she was merely the one to point out how Hildi had big eyes for Django. Thus giving Stephen a reason to suspect foul play between the trio. If Miss Lara had kept her big mouth shut I believe their scheme would've gone unnoticed and nobody had to die.