I'm kinda going through the same thing with my oldest. She'll do something wrong - & usually SHE does start it - then when she gets into trouble she tells me how much she hates me, how much we hate her & how much she hates herself.
One night she actually said that she wanted to die. That was one of the hardest things to hear. It was said for attention only. She threw a tanty about staying over at my brothers place & was sent to bed when we got home. When she wasn't asleep 2 hours later & was still screaming & kicking the walls I turned off her light. I told her I'd heard enough & that it is time for bed as she had school the next day & it was nearly 11pm.
Thats when she yelled out 'I just want to die'.
I didn't know what to do. I sat in my room & cried for a bit, then went back in, told her again its time for bed, that I love her & goodnight.
I walked out & 10 minutes of yelling later & she was asleep.
I know how guilty you are feeling, but you did the right thing. DD is seeing a psycologist for that & alot of other reasons. All I can really do is just try to reassure her that I do love her & try not to react when she tells me or DD2 how much she hates us.