The Kindly Man is the Faceless Man whose role is merged with Jaqen H’ghar in the television series.
In the novels, Arya Stark encounters Jaqen H’ghar earlier, first among prisoners traveling toward the Night’s Watch and later at Harrenhal. In Braavos, however, her principal instructor at the House of Black and White is a different Faceless Man whom she calls the Kindly Man because of his gentle expression.
Game of Thrones simplifies this arrangement by making Jaqen Arya’s Braavosi guide and teacher. The series does not introduce the Kindly Man as a separate named character, so Jaqen effectively absorbs that character’s narrative function. The actor Tom Wlaschiha’s recurring Jaqen persona reinforces the merger.
The Waif is not the answer: she is another Faceless Man who trains Arya and tests her loyalty. Syrio Forel is also separate; he is Arya’s former water-dancing instructor from Braavos, not a member of the House of Black and White shown in that role.