What does Peter admire about Anne?

I could tell you from the book but I'm not sure if the play version is different. You seem to be referring to the play.Give me a little while to find the script and I'll get back to you.

Peter is very fond of Anne. We get the sense he is developing romantic love for her.

Peter: " I think you’re just fine . . . What I want to say . . . if it wasn’t for you around here, I don’t know. What I mean . . . "

Peter admires Anne's smile and loves the dimples which she gets on her cheeks while she laughs. So, he says to her " Do laugh, Anne " very often.