"I'll notice you, dahling," said Flora, putting a posh, actressy, mock-admiring voice.
"And my couplets, dahling. Don't forget them," I agreed, in rich, actorly tones.
"How could I? High spot of the play.
Today is a tree, tomorrow Hamlet, eh what?"
We were doing our daft old routine. But our hearts weren't really in it.
…"eh what?" is part of the "daft old routine" that he refers to, where they are talking in exaggeratedly "posh" voices. It's an old phrase that means something like "I'm sure" or "don't you agree?". It's only used now in frivolous or playful contexts like this.
not a teacher