Hey OP - god do I recognise this!
A bit of background: my DP has a DS11, EOW and 2 afternoons a week.
DP does EVERYTHING for him – helps him get dressed in the mornings, fetches everything for him (will get up from working in his office to get DSS a glass of water, while DSS is playing Xbox). DSS doesn't pick up after himself, doesn't help lay the table for meals, doesn't help carry shopping in the from the car – doesn't even carry his OWN stuff in from the car. No bedtime, no house rules...
I also struggled HUGELY with all of this. Infantilisation, undermining his development, pandering to every need and desire, treating him like a little Emperor... I used to have to take myself out of the house for a walk and a cry (TBF, probably not just about OTT parenting, likely also linked to my own issues after having a pretty abusive and painful childhood, seeing DSS so intensely catered for and loved is sometimes tough).
Having said all that, DSS is a SUPER sweet and lovely kid, and none of this is his fault. It's the effects of Disney Dad-ing and it's not how I'd want to raise my kids, but it is what it is.
A year in, here's what I've learned works for me:
- Go do wonderful things for yourself, on your own, when DSD is around. It's important for them to have 1-1 time, and it's better for your mental health to take space and not watch the madness when you don't have to. And you don't have to! It's early days, you don't have to be around all the time. There's no rush to get to happy-blended-unit. So enjoy it. Go to a yoga class, go out with friends, take yourself for a walk or a massage or whatever floats your boat. Their Daddy-daughter-time can be your self-celebration time. If he's into having a tantrumming 5 year old around, let him. You don't have to be a part of it – and not being a part of it doesn't have to be resentful or borne of annoyance. Just gracefully bow out, blow them both enormous kisses and leave them to it.
- Your boundaries with DSD are yours - and they can be different from your DP's. DSS knows that if he and I are out doing the food shop, just us two, he's expected to help bring the shopping in. He's asked me a grand total of once to fetch him a glass of water from the kitchen when we were both on the couch. I burst out laughing and tickled him, asking what happened to his legs, and he's never done it again since. He knows what I expect of him is different to what his Dad does, and that's all cool. We're making our own groove. In your situation, you may have to tread carefully if your DP won't back you up on your boundaries with consequences, but you may also need to draw some lines in the sand with DP about what you will and will not stand for, when it comes to YOU. You don't get to say how he raises his daughter, but you do get to say how it impacts on you.
- Know where your frustration has its roots. When I asked myself why this was so frustrating for me, I realised it wasn't really about DSS not being raised right. My frustration came from feeling like I lost my DP to someone else whenever DSS was around, and it's the first time I've ever had to share a partner's love with someone else. I felt swept aside by the force of this little person who would always come first in my DP's life, and it was a brand-new and uncomfortable feeling.
Hence point 1 above becoming so important for me - doing LOVELY things for myself while DSS is around - not because DP DOESN'T love me to death, but because it takes the edge off these new emotions I'm still finding my way through.
Give yourself time, OP. Give all three of you time. If he's been single for a while before you, he's not had to make adjustments for the impact of his parenting on a third person before. Neither has she. It's going to take a minute 
I know it's frustrating, and I know the urgency to have it all worked out quickly. I've honestly felt it so hard over the past year. But you will all find your own grooves.