Who is responsible for drug overdose in an assisted living facility?

Whatever MD prescribed the medication should have also ordered regular labs.

AlvaDeer: She almost died but is recovering nicely. The home administers the meds. My sister informed us of the lithium toxicity levels but refuses to answer our questions as to how this could have happened. (She does inform us of a few things (such as almost dying from lithium toxicity)).

The MD in charge of her care should have checked on her lithium levels.
Were you told that her death was SOLELY due to lithium levels?
If so a complaint to the medical board about the doctor caring for your mother is not out of line.
If you are thinking of suing I can explain to you why you should not. There is no recovery really in the very elderly and lawyers will not take suits they cannot recover anything on. The only law suits that recover in malpractice these days are suits involving permanent ongoing injury with life care needed in someone young and well employed with family to support.
There is no reason you cannot see a personal injury lawyer if you are POA, but you are not, your sister is, and doesn't wish to discuss it. Given she will NOT discuss it I do wonder how you got the information and from whom as sounds as though they went beyond what they should have under the "rules"

The nursing home where mom did rehab gave her a double dose of meds. I was really upset. Not as big of a problem as you have but I reported it to the head nurse and the social worker. They did address it.

The LPN admitted it. She told mom to take two doses of her Parkinson’s meds so she wouldn’t have to come back. Crazy, huh?

The facility is! Her Dr should have been testing her levels every three months. It is his/her job and he/she are the ones with the knowledge of lithium and it's effects. Remember a Dr has to document everything they prescribe and all test and procedures. A medical chart is a legal document.

Was the staff giving her the meds or was your mom taking them herself? It may be the doctor/facility since they're the ones who were supposed to monitor her levels. Your sis the PoA would be the logical one to follow-up on this. Not sure if you have legal recourse, but you might. You need advice from an attorney, but not sure what it is you want as an outcome: moving her to a better facility? You may need to pursue guardianship/conservatorship if you think your sis and facility are being negligent. I've heard that if people fight too much over someone, the court may decide to give guardianship to the county instead.