Luton have made their ninth summer signing following the arrival of winger Danny Green.
Notts County have signed Togo international Mawouna Amevor on a two-year deal.
Meyler, 23, has been on loan at the KC Stadium from Sunderland for the past few weeks and signs a three-and-a-half-year deal.
Manchester United's Brady, 20, pens a contract of the same length following a successful loan spell.
The Republic of Ireland international has been in impressive form during his stay on Humberside with the left wing-back, scoring once in 12 games.