Praying at the altar of the Natural Rate
So first the definition. Yes, structural is the antipode of cyclical (well duh), but it is by no means something that cannot be affected (cured if you think of unemployment as a disease) by demand. The only reason to think that the trend cannot be affected by demand is because Krugman believes (based on faith, since logic and evidence are against it) on the Natural Rate. Output is supply constrained in the long run, in his view. Actually, you can bring the average rate of unemployment (the trend or structural one, not the fluctuations) down by expanding demand. During the Golden Age of capitalism, average (structural) unemployment was lower, since there was a macroeconomic regime in place that allowed for consistent demand pressure and higher rates of growth. It involved capital controls and on average low rates of interest, high government spending in defense and infrastructure (in the case of the US), and increases in wages with productivity, so workers could expand consumption.
By the way, his lack of appreciation for Structural theories of inflation is also based on this obsession with the Natural Rate. For him all inflation is caused by excess demand (it turns out that Krugman is a Monetarist). Latin American structuralists pointed out that the changes in the structure of production created cost push causes for inflation. Logic and evidence suggest that they were right. By the way, same reason you shouldn't, as Krugman isn't, afraid about inflation in the US right now. Not close to full employment, and bargaining power of labor class at all time low, with no wage cost pressures. For a slightly longer discussion see this entry on the meaning of structuralist macroeconomics, and this one on the evolution of Latin American Structuralism which is still around (yep, the news of our demise are greatly exaggerated).
PS: I believe that you cannot understand inflation unless you have some structuralist view of it. For more go here.