Once the trust is gone....it's gone. Protect yourself and manage around the lies. I haven't found that there's huge corrections in some of these maladaptive behaviors in SKIDs from childhood into adulthood- it usually carries through IMHO. 

Why does a 9 year old girl have snap chat stories??! DH needs to confront BM about internet usage etc. This is dangerous. He needs to go to BM about why she isn't in school. Her behavior is being taught by BM. 

I would open and close any interface with either of them with "So, are you lying?" and "Was that a lie?"  Every interface.

Push and you will eventually get a tearful "But I'm not lying this time."  Which is when you go in on a feeding frenzy of "Lie once, and you are lying every time."  

If your DH so much as gives you a sideways glance rip off his nads publically. "Your children are liars and I will not tolerate it. So, fix them or keep them the hell out of our lives. Your choice."

First of all EVERYONE lies. 
 

Now the extent to which people lie is the issue. 
 

People who pathologically lie, lie frequently, lie about any/everything, lie as a means of control/manipulation are more of a problem. 
 

Most kids will tell basic lies to get their way or to get out of trouble. 
 

You already know those kids be lying....they learned it from their parents....so instead of expecting them to change if their parents don't care is futile. Hopefully the bio parents can get on board with you about decreasing their lying especially to the people who feed, clothe, and take care of them 
 

If they keep lying at this rate then Just move accordingly. Like don't be one-on-one with them so they can't lie and say you abused or neglected them. Etc. 

Same reason I gave up talking and asking for any accountability of SDs. "I don't know" or silence was always the answer when I couldn't find something in common areas, things were broken, etc. So I installed locks, put nice things away forever (I should get some of those out now, to watch DH closely though with my crystal), and just stopped fixing things (DH then can figure it out if "no one" did it). 

I think if you just withdraw from it it will help. Don't ask, don't look, don't bring it up.

My SO's daughters are pathological liars. When i realized, i stopped believing anything they said, stopped trying to have a relationship with them based on trust and respect, and made it superficial only. I stopped fussing at SO about them, but it's easy for me because we don't live together. When he would tell me about some inconsistency, maybe i would say "Oh? But i thought you told me SD said x...." Just, like i was mildly confused but not upset or anything. He had no choice but to say "Well, that's not what ended up happening." He realizes now that anything that comes out of their mouths is as likely to be a lie as the truth. But if I had been the one to push, you know how it would have gone. 

There's not much you can do.  DH must understand that BM is the problem, not you.  His idea of a "Hapoy Famiky " is gone jts doesn't exist anymore.  He can have one with you and your bio kids but not his first kids. You must disengage. Don't bother with SK,  let DH do all the cooking and cleaning after them.  If you can't have a say, you are not involved by his choice.

Trust is gone with SD13. The lies she tells aren't white lies, they are manipulative. She says outright lies about people to gain sympathy. Years ago when I was 100% involved, she lied about me to BM. I don't see a situation when SD13 could ever really gain my trust back.

She has rejected DS3. It used to hurt my feelings, but now I view it as a blessing in disguise. They aren't ever alone together so it's nothing I have to worry about. I don't have to think about her qualities that could potentially rub off on him. DS3 doesn't even know what an involved "sister" looks like, so (as of now anyway) his feelings aren't hurt when SD13 doesn't play with him... because she NEVER has. And I am finally okay with that. 

I've experienced that also. My SS24 and SS19 were lying from the day I met them. SS24 was 12 and he lied and acted like he was happy I was with DH, also acted like he was happy when I got pregnant and I found out later on he wasn't. He did the same thing your SD is doing, where he would lie about where he was, and DH never really cared or bothered to confirm. He also would just sweet talk DH (hey dad! how was your day? I want to be just like you!) and I feel like those words were enough to have DH never really want to question his son.

When they both began smoking, I told DH what my thoughts were but he was in big time denial. Now...they both struggle with an out of control marijuana habit. 

Fast forward 10+ years and now....SS24 struggles financially and with responsibility. So does SS19. SS24 never updated his checking address which we have asked him to do multiple times so they send letters when he overdrafts. He recently started doing this again (looks like he sends himself cashapp and overdraws his account when he is broke). It just makes me sad for them, because due to DH's poor parenting, they're the ones that are going to struggle with the results of that. And now that SS24 is a dad, that won't bode well for his son either.  

They come by it honestly. BM is sneaky and manipulative but she is also transparent and stupid so it's not hard to see right through everything she does.

SD24 is especially passive about her lying. For example, SD has no regard for DH's time. She will knowingly pick a time that he is unavailable to forward screenshots of texts that she needs his help sorting out, such as health insurance info. When he texts her back something like "now is not a good time, let's talk later" then her response will be something like "I'm sorry I received these texts from the health insurance company" rather than "I'm sorry for forwarding texts to you when I knew it was a bad time to contact you." The way she twists herself into a pretzel to avoid taking accountability is maddening to me.