Both my SD and SS went through stages where basically everything that came out of their mouths was a lie - and they were good at it. Really really convincing. Part of it was the age they were at, part of it was they were telling their Mother what they thought she wanted to hear. Part of it I think was to build themselves up, not understanding that when they were caught out, and eventually they always were, lying always made the situation much worse than it may have been originally.
Eventually I got to the point where I just couldn't be bothered, and told them so. I pointed out that because of the constant lies, even when they were telling the truth, they would not be believed, and it would come back and bite them in the bum. And it did to a certain extent. Having said that, SD is still a very good liar, and still hasn't learnt that if you lie, eventually, somewhere, somehow, you get found out, and people loose respect and trust for you.
I don't know how old your SD is, but I'm assuming old enough to know right from wrong. Therefore, I'd be blunt with her and say that if she has to lie, take it somewhere else. You're over it and actually although you love her, it's making her really hard to like at the moment. Then I'd distance myself as much as possible. She'll get the message.
Your DP knows the truth and so do you, and so does your DD. The only one who is making herself look bad is DD. Let her. That's a decision she is making every time she lies. So be it. Actually, I'd point that out to DD as well, see if it sinks in.
As for what XP believes, who cares? That's her problem, not yours. We all want to be liked, especially if the dislike is unfounded and based on lies, but at the end of the day, you can't change it, so let it go.
You can't make your DP take action over it. Again, let it go.
Eventually, the truth will come out and you will be vindicated. Until then, hold your head up high, practice distancing yourself a little and leave her to it.