So sorry to hear of your situation. It's difficult to face a relative going through a hard time.
First of all, it infuriates me that they did not admit your sister to psych after such a serious attempt (her OD)! That's pretty much unwritten protocol in my hospital- the pt comes in, goes to ICU until they're off the vent, then gets transferred to us. I don't know the whole story, but from what you said, that borders on medical neglect in my book. At that time, your sister would have been totally appropriate for an involuntary admission (I don't know where you live, I am just telling you how it would have been done under IL mental health code).
The criteria for having someone involunatarily committed is "Are they a threat/danger to themselves or someone else at the time of admission?" If the answer is truly yes, and that answer can stand up in court and an MD will be willing to back it, we've got a case. Understandably, if there's not good reason to commit someone, the MDs won't do it because they're legally responsible.
As far as your sister's behavior goes, sounds like she may have quite a bit of an Axis II diagnosis going on (just my opinion, not diagnosing). Those are hard to treat and even more frustrating for the families, so I am sending you lots of support. When you say she's having episodes of drunkenness, is she getting a little drunk once a week or super, blacking out belligerent drunk every night? What I'm saying is, do you think there are enough severe behaviors to have her committed? Honestly, I work as charge RN often and I hear about many pts in the ED who I think are will be admitted right away and then I hear back later that they got sent home after screening because the MDs don't think they can commit them. It's shocking, really. Please don't think I am in ANY WAY discouraging you, I just want you be prepared for what could happen because it does happen frequently (where I work, anyway).
Also, the long term hospitalization- in IL once you're admitted, you're there until the MD discharges you. It might be a day, it might be 3 months. There is no set period and we make sure to tell pts that upfront. If the pt is involuntary, they will be going to court hearings on the floor and the MD and nurse documentation have to maintain good reason for that pt to still be here. If that can't be found in the chart by the judge, usually the pt gets to sign a voluntary which usually means they sign a 5-day notice as well, or the MD has to release that pt. Sorry I can't offer you better advice on the long-term hold part, we don't have a set 24-72 hours for our pts. You also said that your sister doesn't have health insurance, but I would say 90% of our pts either have Medicaid, Medicare, or are self pay.
Let us know how it goes. We are thinking about you.