Elfhelm's text reads:
While Elfhelm is ready, he gains:
"Response: After your threat is raised as the result of questing unsuccessfully, or by an encounter or quest card effect, reduce your threat by 1."'
Zigil Mineshaft's text reads:
Action: Raise each player's threat by 1 to place 1 progress token on Zigil Mineshaft.
So couldn't you just raise your threat by 1, have Elfhelm lower it by 1, and continue until you have fully explored Zigil Mineshaft?