The Dean's mother Susanna Churton was the bishop's sister. Their father was Edward Churton, Archdeacon of Cleveland. Can't see who he married, but Jane Austen's only sister Cassandra doesn't seem to have married anybody.
One of her brothers changed his name to Knight, but none of them seems to have changed his name to Inge.
Not getting anywhere. Hard to see how the bishop would get it wrong though.