Congrats on becoming a new K-F owner! I have a '51 with hydramatic myself. If mine had a tranny problem, I would probably start looking around for a good used transmission, or even a running parts car with a good trans. These transmissions tend to be almost indestructible other than leaking seals, and most people drive them a limited amount, so a full rebuild might not be necessary to a good used unit. You could probably pull the pan, inspect & clean, replace the fluid, change the seals and go.
What is the problem with your existing trans?