I think the easiest way to define a bicategorical notion of end is to follow the identification in enriched category theory of ends with -weighted limits.
The Yoneda lemma for 2-extranatural transformations shows that for a biprofunctor there is an equivalence . For an arbitrary , we then find that if that limit exists, so that we may write this limit as .
One fact will be useful, and it is easy to show: for pseudofunctors , the category is equivalent to as usual in enriched category theory. This is easiest to see by comparing with .