Why wouldn't Ms Norbury just tell Mr Duvall that Cady was the only person who could've written that?

I am not clear what u mean.
After Cady confessed, Ms Norbury said to her, she should tell the other involved people.

What Cady's punishment was?
To join the mathlectic group.
She didn't wanted be there because the guys were so uncool, that it was social suicide.