why not ask her? She's old enough to understand, and modelling acknowledging you've made some mistakes, wanting to put them right, but accepting it will take time and work from everyone is a good example to set, rather than 'we will wave a magic wand over the past and everything will be perfect from now on without ever discussing what was wrong in the first place.'
Sit her down, tell her what you've said here, acknowledge that you haven't always reacted perfectly (but that her behaviour also has some impact) and ask her how she thinks things could be better - maybe you can agree some household rules 'Mum won't nag me about things like making my bed or my room being a bit messy but I won't ever leave dirty mugs/plates in my room, and will put the hoover over once a week.' Or 'Dad will stop telling me to go to bed at a certain time but I will make sure my phone is always on the landing/in the kitchen/wherever by 10pm.' Amended regarding what the battle points are in your specific household.
Ask her to suggest things she might like to do together - could be going out to do stuff, could be at home (baking, going for a walk etc). Be prepared that she might not want to be 'seen' out and about with you -but that could be a 13 year old thing, not a sign of relationship breakdown thing. Even if she wants to go with other people, not just you, take that for now. Most teens won't say no to going shopping (for things for them!) and a starbucks, as a starting point.
Depending on your tolerance some sort of activity that gets you all working together and interacting/laughing/even being a bit scared is great for bonding - something like go ape or coasteering. As long as you can swim paddleboarding is easy to pick up if you do an intro course on a lake on a calm day. Or if everything active is really a no then maybe something creative instead - soap/candle making workshop/baking class? Basically something completely new and out of your comfort zone so you break out of your established dynamics a tiny bit.