My sister dismissed aide. What is the best way to handle a termination?

The two week rule is a courtesy. Its not a law. I know many people who have gotten fired on the spot and that little pink slip handed to them at the end of the day. Unless you have a contract stating differently, no severance pay is required either.

If the two did not get along, then the aide should not be surprised. Maybe your sister should have done it when she handed the aide her check but she chose the phone. Better than a text. I think it comes with this type of job and aides have to expect it. The aide had the same right. She could have taken her check and told your sister she was not coming back.

This is your sister's business. I would stay out of it and let her hire and fire whomever she pleases. I would not assist her in hiring, nor give opinions about firing.

Why do you feel it cruel when you yourself say that your sister and the aide "did not get along?"
At least your sister paid her before letting her go. That was decent of her.
No one should have to put up with someone that they don't get along with in their OWN house. I'm just saying.

A large proportion of people who get the sack, do so because the employer is not satisfied with them. When that’s clear, sugar-coating it is not credible anyway.

The aide was paid then dismissed.
I have no idea the tone or words used, but the order of events is acceptable.

While giving 2 weeks notice used to be the norm for jobs (in my day), for such a personal job, it would be a sour 2 weeks for both sides to serve out. Paying 2 extra weeks may have been nice but probably not enforceable without a contract.

I'd let it go.

Does your sister have a replacement aide? Or does she hold assumptions family or neighbours will fill the void?

(Mine would cancel weekend shifts expecting family to fill).

Why does your sister need an aid? What reason (if any) did your sister give for firing her? Does she have cognitive issues? It's pretty common for people with dementia to fire aids because they don't like the loss of privacy or don't think they need the help. At least she paid her.