I had the same problem about a month ago. I soldered on a new lead to the glow plug, and it still didn't work. Then I bought a complete wiring harness, and it still didn't work.
I had to change the body so I could get to the plug with a glow igniter. The problem is in the hand held starter unit. My LHS says that a few others have failed the same way. I haven't taken it apart yet, but there is either a problem with that pin connection, or there is a defect on the circuit board that burns out.
Sorry for the bad news, since the advantage of the EZ start was NOT to have to mess with igniters.