Somebody's Sister question

I recently read Somebody's sister by Derek Marlowe and although it was clear that detective Brackett's all assumptions were not correct, I did not understand the ending. The murderer used to talk in a way saying "...baby" and even in the end the officers find out that Plomer is the murderer then what was the last page about Kemble saying something about palm trees and ...baby. It made me so confused, was he the real murderer or what? And what was the last line in which Brackett is saying why don't you do it...:/