Open the jaws of the current, keyless chuck as wide as possible and remove the screw that goes from inside the chuck into the drill body. The screw removes with a clockwise turn.
Tighten the keyless drill chuck onto the short arm of the hex wrench. The chuck is torqued onto the drive shaft of the drill, and the wrench will be used as a lever to break the chuck loose so it can be unscrewed from the drill body.
Hold the drill firmly on a solid surface with the chuck extending out over the edge of the surface and the long side of the hex wrench in a horizontal position. Use a mallet to strike the hex wrench, which turns the chuck in a counter-clockwise direction. The drill motor spins freely, so it may take several attempts at striking the hex wrench to "break" the torque on the chuck so it can be unscrewed from the drill body.
Screw the replacement, keyed chuck on the threaded shaft of the drill body. Tighten the chuck as much as possible by hand, even if the drill motor rotates as you turn the chuck.
Reinstall the screw through the inside of the chuck into the drill shank, securing the new chuck to the drill.