I have some advice.
First: about the bathing situation. She is how old? 12? Explain to her how WONDERFUL it is to take a nice relaxing BATH!!! Soaking in a tub, relaxing, how women pamper themselves with a good bath every so often and its great. Then put her in a tub that has bubbles, hence the soap is already in there!!
For the Son, Get liquid soap that is fun. Its colored, or foamy, something he will want to put on himself. They have lots of fun stuff out there. OR have DH walk in the bathroom and put soap in his hand (liquid). He can do it with out sounding mean.
Put a laundry basket in the bathroom and have them put thier clothes in it right away. As soon as they leave the bathroom empty it and ask for any clothes that are missing that second.
It sounds like thier mother didnt raise them to be very independant. Well, what can you do, that is thier mother and whether your husband and her were still together or they are living with her, they were going to be raised by her the same way. You DH only goes so far with the parenting and thats the father he would have been regardless.
So because they are not independant you cant just ask them or tell them to change thier underwear, you have to get more involved in making sure it gets done without making them feel bad about it. After all its not THIER fault, they are a product of thier rasing. So treat them as such. Like they are special. Is it "fair" that his kids need more special attention than yours? No, but thats how it is. Stuff they do at thier home that to them is normal and right is apparently wrong at your house. they dont get that. So treat them with kid gloves and be nice about it instead of punishing them for bad hygiene, do the positive reinforcement tactic, or look for something else that works.
My SD has a book from American Girl that is called "The Care and Keeping of You". She is too young for the back of the book that talks about periods, but the begining of the book is all about "how to" brush your teeth the correct way, why you should wash your hair after swimming, how to correctly clean your ears, what type of clothes to wear and when. Its like a girls body users manual. I recommend it. It is complete with pictures (drawings) and explinations and side bar comments. It even says how to brush your hair (with a wide tooth comb if its wet, a large brush if its dry).
the other issue is her lying about how your kids are mean and how you are mean. Well she probably REALLY does think you are mean. Maybe your kids dont talk about how they have so much more, but maybe she feels like it. My stepsister Samantha is 30 years old now and she still makes snide remarks about how we had a better life than her yadda yadda, and she really does think thats how it went down. Even though we grew up in the same house, and my full blood sister and I lived with her because neither of our parents wanted us (lived with aunt, so I guess Samantha is supposed to be my cousin by blood, but sister by love) she thinks that somehow we got more than her.
So maybe when she is there she sees simple comparisons in her head and she feels like they are rubbing it in her face when really they are just living thier life as children raised to be independant and have more. ie: thier toys arent broken all the time, thier room is cleaner. Thier clothes dont stink!!! Life is better and simply breathing in her general area makes her feel that way. The funny thing about kids is you would think they woudl be happy in a life without structure but the truth is they crave it, which I'm sure you know having raised 3 daughters yourself.
As far as her not liking you. Well I'm going to try not too be to harsh on this one. Your putting a tape recorder in the room to catch her in lies, your going with her father to confront her about the lies, and your fighting with her father about how she is bad and trying to get him to see that. You know as well as I do, feelings you have on the inside show on the outside. So she knew you thought she was bad even before you started proving it to her. Shes certainly not goign to talk about how wonderful you are if your trying to prove how bad she is all the time. Expecially if she knows that when he father tries to handle the situation you turn around and tell him that he didnt do a good enough job. If she knows that, then your screwed.