The Republic of Ireland internationals, who have both agreed three-and-a-half year deals, have been on loan at the KC Stadium since November.
Meyler, 23, has scored three goals in 10 appearances during his loan spell from Sunderland.
Brady, 20, has played 12 games this term after also spending last season on loan from Manchester United.
Hull City have completed the signing of midfielder David Meyler and winger Robbie Brady for undisclosed fees.