I'm flabbergasted that neither parent has stepped up to figure out what is going on. Taking pills and putting bleach in the mouth ARE suicide attempts, and those "less lethal" forms fall more in line with attempts made by women.

SD needs therapy, and yes, she needs a babysitter. When YSK was actively suicidal, we didn't leave them alone except to sleep. We did knife counts. We made sure they didn't have access to medication. We kept cleaning products in our bedroom. I think the first time we left them alone was to go to dinner, and that was only after their therapist told us they were stable enough.

This isn't your responsibility to address, but you could end up being at fault if SD does injure or kill herself when you knew, as an adult in the household, that she had made suicide attempts. If you want to protect yourself, you may need to consider moving out if your DH or BM doesn't step up to actively help SD. It's one thing to not know and have it happen, or for her to be actively in treatment and it happen. It's a whole other ballpark of willful neglect when SD has harmed herself multiple times and no adults stepped up to do anything about it.

Make it abundantly clear to your DH that SD needs to get help and a keeper during the day or you are gone. If SD is taking it all, a few therapy sessions, locked down social media, and a babysitter every day will get her to stop (though I contend any child who is hurting themselves or pretending to hurt themselves for attention still needs therapy and help). If she is serious, though, but just hasn't been able to fully commit (which happens more than you'd think), it might actually help her become functional. Ignoring it, though, spells legal disaster for you even though you're not the parent.

This is a kid who was told by her mother that she didn't want her ... Please take these suicide attempts seriously because sooner or later she will succeed. Bleach can most definitely kill. She needs therapy and she needs it like yesterday. Suicide attempts are a cry for help. Please get her that help.

As for it stemming from her social media, can't you see any other reasons she'd want to take her life? OTOH, if she has unfettered access to the Internet, she will eventually find an efficient way to top herself. Don't let it come to that. Please.

This is less about you not being alone with her than her not being alone at all. No more leaving her alone when you want to go out. She must have company. And she absolutely must have therapy. Has she ever had a mental health assessment? 

Please take these attempts seriously. As a mental health professional, I have seen several attention suicide attempts turn lethal. One in particular always stands out in my mind - he took his roommates pills (only 7 pills) that he didn't know anything about & killed off his entire imune system. This was a 32 year old adult not a child. It took over three months of torture for him to die. All the while, begging not to let him die. He eventually died from sepsis which was extremely painful. So much so, it still haunts me almost 30 years later. Had SD tried antifreeze instead of bleach she would likely be gone. 

This child needs serious professional help. No one that threatens suicide should be left alone. Hospitals even pay for 1:1 staffing for suicidal patients. 

It is unfair (and unsafe!) for this unstable SD to be left at home with you while you're working.  You cannot supervise her while you're working, nor should that be your responsibility.  But clearly she cannot be left alone.  Even though it would not be at all your fault, can you imagine the guilt you would feel if this child attempted to, or even worse, DID, kill herself while she was in the house with you while you were working?

This is the proverbial hill to die on.  Tell your husband in no uncertain terms that he needs to make other arrangements for his daughter.  If he does not, give some serious thought to why he is not willing to consider and prioritize YOUR feelings and desires.  A good partner would.  And then get out of the house.  Can you work elsewhere?

It's good she's admitted, that should happen every time she attempts something. This might come across as snarky but I am dead serious. Whenever a kid has a suicide attempt you have to make the biggest deal about it possible take it as seriously as if you had found a gun in their hand.

There is an attention seeking aspect to suicide attempts but at its root is a cry for help. So you give them that attention you make a huge deal about it you start treating them like they can't be left alone take them to mental health professionals at least once a week and do all of the follow-up. In other words make it a huge deal and be super annoying about it. They cannot get mad at you for taking it seriously and if they ever start saying things like "oh it wasn't that big of a deal" then you know that they are probably not going to try it again because you made your point. On the other hand if they are very seriously troubled taking it to the extreme is the right thing to do to get them help and dig out of the hole. 

Usually after being admitted a couple times there are no more casual attempts. 

As a parent I would go all in on aggressively addressing the suicidal tendencies or behaviors.  If a kid would kill themselves I would be destroyed but I would do everything in my power to address the risk assertively for as long as it would take to get them "well", or... they succeded in their efforts.

Heaven forbid.

They would know I cared, was taking action, and that their tendencies would be met with full bore intervention including institutionalization, full time professional oversight including while using the toilets, baths, showers,  a home stripped of any and all risky items that might be used for suicide attempts (knowing fill well that if htey want to, they will find a way).  They would live life under the supreme hairy eyeball for as long as it took to support them through those tendencies.

Fortunately, so far anyway, our kid seems reasonably safe from this risk though I do occassionally grow concerned based on his extreme introversion,  cyclical moods, and not irregular long periods without communication.

I truly hope that my default state of extreme optimism is not hiding a risk.  The thought is alarming.

I think it's best to know that unfortunately it doesn't get better from here. The fear will always be there for you regarding where she is and if she succeeds the guilt eats you up. Regardless of what the parents are doing I would be advocating for this child to get her every bit of help or whatever it is she needs. When others aren't fulfilling their role or duties someone has to step in. It's very hard to turn your back knowing the parents won't step up. This would be the hill I would die on because for me, it's morally incorrect how the parents aren't stepping in. That could also be driven by fear but they also need help navigating through this. 
 

I'd believe anyone who says or acts suicidal. The consequences are to catastrophic not too.

She doesn't need to convince you that she's suicidal, she only needs to convince you that she needs help and I think you've acknowledged that.