Ok - here is a tried and tested way of dealing with this (from a professional kid/teen wrangler).
Check if she is being bullied at school - always a good place to start.
Ruke 1: keep calm at all costs. If she has a tantrum, or is rude, or smashes something, don't get angry, threaten dire punishments or lose control. If you feel as if you are going to start yelling, remember something that you have to do in another room and count to 10 (or 100).
Rule 2: no bickering with DP. Kids really get upset by this and can act out.
If she breaks something - if it's hers, it will not be replaced and if it it someone elses, she will pay to get it fixed/replaced (pocket money or chores).
Start with her bedroom. OK, she's a 'big girl' now, time for a big girl clearout! Go through all the toys and tat, and box up the things that are broken and not age appropriate (she can keep a shoe box, or specified number of 'special' things). The stuff to go can donated to a charity of ebayed (or you pretend to and throw/give to charity) with the money going to her as pocket money which she 'earns back' by keeing the room tidy. Maybe 50p a week after a 'check' to make sure that clothes and toys are put away, books are tidy and bed is made. Make it sound like a fantastic activity - all very positive and about her.
This helps her feel a bit more in control of her own space and also gives her a calm place to chill.
Next - decide what 'rules' will be put in place. Write them up! What positive behaviour do you want? Bed time, helping about the house (emptying wastepaper basket, feeding the goldfish, taking her dinner plate to the sink, finishing her meals...). There will be a 'fine' for not following the rules - a penny or half penny which you dutifully wite down. She can 'earn' pennies by being polite, getting dressed for school on time, folding her clothes, making her bed etc, or just a general 'reward' eg she gives a toddler back their toy that they dropped in a store without prompting.
Going to bed without a squeek earns pennies too. The 'bad' stuff is fined:
You 'can you please take this to daddy?'
DD 'no - won't!'
You 'ok, that's rude and its a fine... 1p'
DD 'nooooooooooooo'
You '2p, (pause for 5 seconds as long as the tantrum continues) 3p...pause...ok, slamming the door is another 10p...I can do this alllll day!'
It goes on like this - the first few times will be doozies but the key is for you to be calm and not get upset/angry/emotional. If anything, be slightly bemused 'such a big girl...oh well, another penny fine for being rude...'
If the has a major tantrum, ignore her. Take yourself away or ask her to go to her room as she is disturbing your nice cup of tea with DP. Look lovingly at your DP and ask of he would get you a nice bikky to go with your cuppa. He nicely says 'rich tea or hobnod, dearest?'. This really confuses the child as it is not the reaction she wants! She wants sparks, drama, emotion, then the whole comedown, tears, apologies and cuddles. Don't feed the flames - ignore the tantrum. Focus on something else completely - you are not prepared to talk to her until she has simmered down, and that's that. Obviously, as long as she isn't in danger.
Balance with loads of praise - 'you got dressed for school all by yourself and I didn't have to ask you! Great!!!'
You will be in control and kids understand pennies! You need a calm home - you are not well! She also needs a calm home - I know you know this, but the years ahead will be tough for her with you being ill and maybe not being around. If you have a calm, 'safe' home then you can deal better with whatever life throws at you.
Make sure that you and your DP go through the ground rules when you are alone. Agree to work as a team (kids can see where there's a chink in the wall and explpit that). Be consistent, and loving to the child - but firm, calm and follow the rules.
You are the parent and need to be in control. She will kick against any new regime and expect behaviour to flare up but it will calm down, I promise. You can do this!
Being tough will not work, nor will being too soft. Yes, tell her you love her to pieces but that when she has a tantrum you are confused because you don't understand why. She probably won't be able to explain why she does this. Tell her that these rules are for the family, and to help her be a big girl.
Each week, check the earnings and fines. You can 'fake' results so that she earns a few pennies/pounds each week and praise her to high heaven. She won't remember the scores anyway. 'What??? We owe you 97p this week! Goodness, you have been on great form! Maybe next week, mummy and daddy will win ths race and you will owe us some pennies!!'.
Work as a team with your DP. It will be worth it.