I had a glow plug (GP) controller fail by remaining 'on' constantly; I don't know why one couldn't fail as you describe.
My book says the controller senses ambient air temp in order to decide how long to cycle the GPs. I don't know if the sensor in your case is incorporated into the control box or is somewhere at the end of a wire. A parts schematic for your machine would be useful.
If you had access to power I'd suggest using a hair dryer pointed up under the dash just to see if the heated air makes something change. Along the same lines, with power, a lower radiator hose heater could make your life better. Not trying to beat a dead machine (with an extension cord you don't have) but have read here that cold engines have sometimes been started by pointing a hair dryer into the air intake when starting.
I don't have a Workshop Manual handy but I think I remember mine has a time-vs-temperature chart showing how long the GPs cycle based on air temp.
If you start replacing parts, try replacing one component (you may have a separate 'timer' and 'controller') at the time until you figure out which is failed. Try to get approval from your dealer to return the unused pieces. My controller was about $75 couple years ago.
If you have the ability, and the machine allows it, you can pull start it. If it gets fuel and air and you can build compression sooner or later it'll light off. I realize it's a different lash-up but I can start our old Army 2.5 by pulling it about 5-ft.
Other than waiting for Spring, I'm curious how you proceed and what you determine the problem is. Please post back that we may all learn.
