When i use the counter "end the run", have the runner in all cases to stop the run?

I think so.

Or can he pay the costs from an icebreaker?

The ability of Nisei Mk II ends the run. There's nothing the runner can do about it. Certainly not with an icebreaker, because the effect does not originate from a piece of ice.

Yeah, but make him run through all of the ice first so at least it costs him money.