Should these accounts be included in Probate?

Sorry about your dad.  When you read about joint accounts, including here at The Forum, there is often an assumption made that a joint account with rights of survivorship will pass to the surviving co-owner, outside of the probate process.  And usually they do.  The co-owner goes to the bank after the death of the other co-owner, and gets the money.  In that case, the co-owner--you--can do whatever she wants with the money.

However, that assumption can be challenged by a beneficiary of the decedent's will.  When such a challenge is made, in trying to determine whether a joint account should be considered part of the probate estate, the court will consider several issues, including the decedent's intent.  The court will try to determine if your father opened the account and/or put your name on it for his convenience, or did he do it to allow you, the co-owner, to have access to the money for any purpose.  In this case, it sounds like you put your name on the account, in your capacity as your dad's power of attorney, after the account was opened. These questions may not be very important if you and your sister are the only beneficiaries.  But if there are other beneficiaries who might possibly object to your handling of the money, then you need to think about it. 
 
Courts also consider who paid the taxes.  If, as in this case, the income taxes were paid by the decedent, that indicates the account was really the property of the decedent and, accordingly, should remain in the probate estate.  Similarly, if the surviving co-owner never made withdrawals from the account for her own use, then the court might conclude that it belongs to the estate.  Finally, the court may consider who kept the passbook for the account, although passbook accounts are not very common anymore.  Hope that helps.