That is definitely a project.  It looks to me like it was fed a steady diet of high iron content water over all those years.

I had a later Imperial model for many years.  If the timer and buttons work the same on yours, here's my guess as to how you'd use them:  Pushing the "Full Cycle" button will require you to rotate the timer knob to the thick line between "Prewash" and "Wash" sections in order to start the machine.  Pushing "Rinse Hold" means you will have to rotate the timer around to the thick line between "Wash" and "Rinse" sections.  I think "Soak" is just an outdated term for "Pots & Pans."  Push that button and you probably just have to give the timer knob one click from the "Off" position to start the machine, and you would probably fill both detergent cups only when using this option.  My guess is that the machine knows enough not to apply heat as it advances through the "Dry" cycle when you select "Rinse Hold" option.  The fact that you don't have an air-dry option is all the indication you need that this series was produced prior to the 1973 energy shortage.

For plastisol touch-up on the racks, you may have to do some searching to find it in blue.  I found Maytag brand (apparently no relation), but it's a very blue blue.  These days, hardware and appliance parts stores only carry plastisol touch-up in gray or white.  My Maytag stuff came from an old packed-to-the-ceiling appliance parts shop with loads of old inventory.

I never ever have seen a build up like that.

IMO, the Wash Arm support looks OK. The mineral deposit is scary.

Next thing I would do is check the Intake/ Sediment screen on the Fill Valve. It's probably 98% plugged.  Sounds like and looks like it's starving for water. You may have to replace the valve just by the looks of the tank deposits. But you never know until you inspect that.

Underside of the wash arm... Not too bad.  If you can stop the leak, after you clean the line strainer and actually get water in the tank, some sort of Lime Scale Remover is definately in order.

But the Experts will chime in shortly.  Especially stevet the Hobart Man.  Hope this helps.

Hey Tim congratulations and welcome to the world of classic KA DWs. This machine is very saveable and parts and racks are all over the place.

This machine is a classic example of someone who pre-rinsed their dishes, How can I tell, well after 40 years of repairing dishwashers one look tells me that there are several different things wrong with this machine. And if someone had actually been putting dirty dishes in this machine they would have not come out clean and they would have called for service LONG ago and gotten it fixed. As Eddie pointed out the machine was likely not getting enough water, this is far more likely a bad valve seal than a clogged inlet screen, but either is possible. The leakage and rust around the dryer unit is mainly from foaming of the wash water [ another usage problem ].

This was the last model before they gave you the option of air-dry, but it is very easy to just disconnect the dry heater and leave the dry blower connected [ you can even wire in a hidden switch so you could switch back and forth if you wanted ]. We used to modify this for customers that wanted to save energy and reduce the damage that the heated dry did to the RR corner of the lower rack.

I was describing the process on my KDI-21.

From the above descriptions, it sounds like the KDI-57 has a rapid advance timer.

If not, then it's like my Thermador THD-3600, which required only pushing the cycle button to start, but then the timer started advancing -- not rapidly -- until it reached the zone where action would begin, by which time the water in the line was no longer hot.