Describing where a Kleisli Triple fits into a Monad ontology I'm trying to map a Kleisli triple onto my existing understanding of Monads. 
I can represent my understanding of Monads like this: (courtesy of Jim Duey's slides at 13)

Could you please point to the part on this diagram where Kleisli triples fit in - or even better - draw another diagram that this diagram can fit into that explains it?

Kleisli triples fit in the diagram exactly where you have "monads". Kleisli triples are equivalent to monads (you might say they are one presentation of monads). 
I infer from the link that you are thinking about this in the context of programming languages? This question/answers might help: What is a monad in FP, in categorical terms?.
