The punishment won't teach her anything she doesn't already know (she's a bad person, she deserves bad treatment). Oh, and the punishment was irrelevant to the crime, so it will be experienced as meanness, not necessity. If you ever feel compelled to 'do' anything punitive about a crime, it needs to at the very least be reparations for the crime.
Confronting her with an opportunity to lie is baiting her to behave badly. Instead, say 'oh, don't argue about it, I know exactly what happened, and I'm concerned about you and what you'll think of yourself if you continue doing this.'
It is likely that the thefts are habits rather than conscious plots to do wrong -- that she has learned so long ago that she 'can't have' that she feels compelled to 'take' whenever the feelings arise.
I would suggest 2 things: be with her 100% of the time (can't steal things when people are watching) and; offer her a great deal more than she needs over and over and over and over and over and over and over again.
She has concluded that there is an insufficiency in the world. Convince her otherwise.