Shimano 3 speed coaster hub cable replacement.
 

Hello,
I have an old Miami Sun tricycle I'm working on. It has a Shimano 3 speed hub with coaster brake. The trike has been stretched and the shifter mounted by the seat rather than on the handle bars. Basically, I tried to shorten the cable length without fully understanding how the shifter works ( I know now). The cable housing has built in ends that attach to the shifter and the hub. One of these ends is now disconnected, because I cut it. How could I go about replacing or fixing this problem? Are these cables still sold?

yes , ask at your LBS ,
in many cases Shimano copied the Sturmey Archer hubs.

kind of a no brainer, make some spare parts interchangeable.. given
Sturmey started making IG hubs 100 years ago, so had a long time to set a precedent.

Your local bike shop should stock a replacement barrel for the end of the cable.
It attaches with it's own nut and bolt.
If you can't find one, let me know. We have hundreds of 'em.