It seems pretty obvious that Law cured his White Lead poisoning with the Op Op Fruit.
I'm guessing Flamingo lets him eat it so he can do the Eternal Youth operation on him, and it won't matter if it kills him since Law is dying soon anyway. But obviously Law betrays him instead.
I don't think Law's sister is really dead either. Oda could have just killed her along with the rest of the family, but instead Oda went out of his way to make sure she died in a way Law couldn't see it happen.