Talk to a counselor, and then have your husband go in with you and hear from the counselor what is appropriate or not. It is very hard on a child when the parents don't draw appropriate boundaries. Believe it or not, it is psychologically distressing to them to be asked (implicitly) to behave like an adult. Your husband should learn this.
I'm going through the same thing! My bf and I live together and prior to me moving in, his daughter slept in his bed. She's 8, he also has a son who is 7. They both would sleep in his bed, more her than his son. On the nights I would stay over the kids usually slept in their own room but now that I live there his daughter always wants to sleep with us.
Furthermore, the bathroom is a whole nother issue. My boyfriend would walk around naked, more often when the kids were there then he did when it was just me! I couldn't believe it. He would walk out in the living room and talk to the kids as if he was fully dressed and talk about the weather or whatever... but it was way too awkward for me. One time the kids were laying on our bed and he layed across diagonal fully naked with his penis flopped out chatting away... once I came in the room he got up and proceeded to the shower. Most of the naked moments are right before he gets into the shower. I told him I think it's strange that he does it and that his daughter stars at his 'area' when he's naked. I also told him that I've never seen my dad naked growing up and that it's not something most men should do in front of their daughter. That conversation happened 8 months ago, and I don't believe he's been naked in front of the kids since. But he will wear his tiny underwear around for ten minutes after he gets out of bed, which is odd because 99% of the time when it's just us he puts his pajama pants and t-shirt on as soon as he wakes up. I know he's not trying to avoid being naked in front of me but he seems to be the opposite when the kids are around. It's weird. He's Dutch and I think they are more open about being naked in front of family...
My thoughts are your husband is trying to please your SD by being the 'fun' parent, he probably feels remorseful for the divorce and how it affected her, not to say the divorce was his idea or not but fathers tend to dwell on the fact their child has to be raised in a split household no matter how crazy his ex may be, it's just the fact that his daughter has two homes instead of one So perhaps since he's not disciplining her it's a defensive mechanism to help him cope with his 'shame'. Men forget that it's possible to set boundaries as a parent and still be loved by their child.
Also, men have a hard time grasping the age of their children. He may still see her as a 4 year old. Perhaps, on the sly you could encourage (make) her take on some 'adult' responsibilities his mind will shift. Like giving her chores to do, setting the table, or help with folding towels, sweeping the floor, dusting furniture. Remind her in front of him she's old enough to do these things and that families work together in keeping the house clean.
Believe me, if you wait for your husband to take action you probably will be waiting for a very long time. He probably needs a little help in the parenting process.
With my boyfriend and his two kids, I almost die when he gives them baby forks and sippy cups. It's as if they're still toddlers. I just ignore it and give them normal forks (small ones) and kid cups. I ask the kids to do me a favor and put the place mats on the table, I just slip in chores and tasks. If they asks for something because they've lazy and can reach the item on the shelf or fridge, I respond by saying you know where it is... but with my boyfriend he'll run & jump for the kids. If he's eating dinner and the one kid asks for ice cream, guess who gets it immediately. If I'm there, I say not until we're all done eating, have respect for those who are still eating and my boyfriend doesn't move.
With the bathroom issue, have her get used to the word privacy. The next time she attempts to go in there say "SD, how about you give your daddy some privacy, I think he would prefer that". If you say it enough for her to understand the definition, she'll get the hit that it's not normal and start to think before she impulsivity walks in there.
Last year my BF's son was obsessed with trying to see me naked, he never did but boy did he try. Whenever he knew i was going to change he would follow me into the room, I would close the door and he would say I'll be in the hallway, you can open the door a little bit... so I started saying that I needed privacy and it worked. After a few times he realized that it was not good for him to come in if I said the word.
Now we all say it. I'm going to the bathroom, I need some privacy, or I have to put my pajamas on and I need some privacy.
So my advice is, just slip in the mild changes with an encouraging tone and I'm sure your husband and SD will adjust without hesitation.