That
Improved New Royal does look exactly like my machine. Seems likely that this machine uses 15x1 needles.
The controller might work and all I need to do is replace the cord. I'm really curious to see what the rheostat looks like and I want to open it up, just trying to find out if there is anything I need careful about. I would hate to damage it.
Here is a better picture of the controller.
I took your suggestion and have found one of these controllers on Worthpoint, but unfortunately it didn't any info that I could search on.