What people generally do is find a donor overdrive transmission assembly (gearbox with the overdrive) and the drive shaft. The transmision and overdrive unit are checked over and serviced or repaired as required and then the unit is installed in the car. Bolt pattern for the flywheel is identical and the rear axle end of the overdrive's drive shaft matches. The various electrical components mounted on the firewall and other places (such as the relay) get installed and properly wired. An overdrive control assembly (with mounting bracket) comes out of a parts car as well and goes into the vehicle.
Consult the transmission, overdrive and electrical sections of the 1951-53 HJ Shop Manual that Kaiser-Frazer put out for more information, diagrams and other details.