There was a good piece of advice above, treat the 10yo as more of your equal than someone you boss around. (Obviously outside of the younger ones' earshot) ask her to help you out with the little ones, you know how they are, etc.
In which ways is she being relied upon too much? If it's something like she has to pack lunches for all of them or do all the vacuuming, perhaps take one of the tasks off of her, specifically telling her that you'll do it for her and that she should go enjoy herself. If the younger kids are required to do much less, especially the 8yo, try to right the balance and point out that it isn't fair to pile everything on the 10yo in front of her.
You know what she's interested in, perhaps get her some activity that she would enjoy for the two of you to do together - but again, as equals. I'm thinking light bribery is nothing to be afraid of, within reason.
Positively reinforce her good behaviour. If you've gone the whole day without arguing, tell her you've had a really nice day with her. If you have a good chat, tell her you enjoyed it and hope to do that again. When she helps you, thank her for it. When she's being provoked (or even not) and is quietly fuming but keeps her mouth shut, praise her for not snapping and treating the others with respect.
I had a 9yo who was very defiant and rude to me, but she loved me at the same time, so if that's the case with you, catch her in those moments when she's more open to you, talk to her about how you like her and want to be her friend and help her, and that it really upsets you when she dismisses you, etc. Ask her if there is something that would make it easier for her to treat you better or what annoys her about the things you do so you could improve.
If she hates you because you just don't click or have a personality clash, speak to her on a more of a grown up level that you want to be treated with respect and then you will treat her with respect, because you reap what you sow, etc.
Pick your battles and if you see that she's wound up already, don't push for silly things. Pick a few things that are absolute musts for you and stick to those.
Again, if she's open to really talking to you, tell her you know that she's in a really tough situation with the mum being sick and all that, and that she's relied upon for so many things, acknowledge that she's doing a great job with it, acknowledge that it isn't fair, offer her to always be ready to listen to her.
Encourage her to express her feelings in healthy ways, by talking to you or her friends, get her a diary to write things down. If she's already using the internet, show her how to set up a private diary online (and explain all about online security, etc), suggest she keeps a diary in Word or something along those lines.
Get her some books that somewhat reflect her life. As far as I remember Jacqueline Wilson has a book about an over-burdened pre-teen with absent parents, and I'm sure there's many more others.
I'm sure your relationship will improve with time. It can also be very rewarding - my rude 9yo is now 18 and we're good friends still.