A lot of people have made a lot of suggestions on the chore issue so I won't say anything about that. I have that issue in my household too, and I don't know that I have the perfect solution.
One thing I see is how you said it's hard to talk to her about it because she reacts. So maybe you could say to her how much you want to get along (prepare her to be favorably disposed to what you have to say), and then explain what she's doing (flying off the handle) and tell her it makes it hard for you to talk to her. Ask how you can talk to her so that the two of you can have a conversation and work out the details. If you just directly present her with the problem you're having this way and ask for her help, maybe you'll get somewhere. There's probably somethig you don't realize that's causing her to over-react, and she may not realize it either.
This worked for me, when I was getting really defensive every time my boyfriend tried to talk to me about a certain thing. Finally he told me, "It makes it really hard for me to talk to you when you react that way" and asked for my help, as I suggested. I immediately wanted to help more than anything, and I was able to improve my behavior. Maybe it'll work for her too.
