The Batman Family: Katherine Kane, The Batwoman

Katherine Kane:

The Batwoman remains one of the more obscure members of the Batman Family. Yet has become somewhat of a cult classic among fans. Katherine Kane inherited a large fortune from the Kane Family in Gotham. Whatever the Wayne's didn't own, the Kane's did. Katherine, who had entered into a relationship with Renee Montoya, (the second Question), was once confronted by her that she didn't have the good in her to do what was right, which prompted Katherine to become the Batwoman after their breakup. For a while she frequently teamed up with Renee Montoya and the First Question before finally meeting the Caped Crusader.

Meeting the Batman:

She wouldn't consider herself part of the Batman family, despite the likeness of their names, because she has had few encounters with him. But there is some history. After Bruce Wayne supposedly dies during the events of Final Crisis, and Dick Grayson become the new Batman, Batwoman became the lead of Detective Comics for a short period of time. When tracking a religious cult within Gotham she meets with the new Batman to discuss the growing threat from within Gotham. Reluctantly he gives the case up to Batwoman and focuses on other problems at the time. She tracks the leader of the Religion of Crime, called The Mad Hatter.
Following Bruce's seeming death the Justice League is ripped in half and she was instrumental in keeping contact between the two new leagues.
Another encounter, (and perhaps the most explicit) is when she is kidnapped by cultists in London. Dick Grayson, as Batman, tracks her down with a pair of British superheroes, (Knight and Squire). They fight their way out of the cult who appears to be followers of Darkseid, (the world destroyer from Apokolips), where they defeat a clone of Bruce Wayne and quite possibly save the world.