Very generally, after someone passes, POA powers terminate and the named executor in a will (if they choose to accept the duties) or the individual that qualifies as administrator will be the one to manage all continuing matters.
Please note that this response is based upon the limited information available in the question. In addition, it is not intended to create an attorney/client relationship and is offered only as general information and not as legal advice.
Some of your statements are a bit confusing. If you (or anyone else) is an agent under a POA that ends at death. After death, the estate is formed and handled by an executor under a will or administrator if there is no will. You say your sister is "named administrator" _ so she either is the NOMINATED executor in a will or you Expect her to be administrator but she can't yet be administrator if your mother is still alive. It is the executor/administrator's job to handle all the duties you mention.
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The duties that you have as your mother's agent under her power of attorney terminate when she passes away. Not only are you not responsible for wrapping up your mother's affairs after she passes, but you have no authority to do so, unless you are the administrator or executor of her estate. Since it seems from your question that your mother has nominated your sister to be the executor of her estate, your sister will be responsible for wrapping up your mother's affairs, if she agrees to act as the executor. If your sister refuses to act as the executor, and your mother has not named an alternate executor, you or anyone else may apply to be appointed as the administrator of your mother's estate.
This answer is for informational purposes only. It is not intended to create an attorney-client relationship and may not be relied upon as legal advice. A careful examination of the facts is necessary before a legal answer may be relied on. You should consult your own attorney before taking or refraining from any legal action.
While your mother is incompentent, you are taking care of her affairs under the Power of Attorney which was granted to you by her before her incompetency. However, this Power of Attorney expires upon your mother's death. You indicate that your sister is named as administrator, but do not say what document names her as administrator. Most likely she is named as an Executrix under your mother's last will and testament. As Executrix, she would have to qualify with the Circuit Court in the jurisdiction where your mother lives (her domicile). Then she would be responsible for settling your mother's estate, such as paying debts, filing taxes, disposing of property, filing papers with the Commissioner of Accounts and distributing money and property to the persons or entities named in the last will and testament. The issue of bank accounts may be different since bank accounts are governed by the signature card signed by the decedent, and the deposits would go to the beneficiary named on the card. If no beneficiary is named on the card the bank account is handled as part of the estate.
The statement above is a general statement of the law and is based upon an incomplete knowledge of the facts of this specific case, and is not intended as specific legal advice. Since specific facts can alter what legal advice is appropriate, always consult with a lawyer for specific legal advice for your situation. In addition, I am not assuming any lawyer - client relationship by giving this general statement of the law.