Just a few ideas to throw at you:

what do you *need* her to do? Anything beyond that, I'd let go. She obviously wants to fight about it, so I'd just not go there. Sometimes I get caught up in making DS do things because I want him to do as I say, instead of taking a step back and thinking about whether it really matters anyway.

Try time in together, rather than time out.

You've mentioned one-on-one time with you already, I know. That one makes a big difference when my son when I can manage it.

Diet and tv time make a difference to my son's behaviour, too.

Get some time out for yourself