Because Pseudsphere is not a closed surface and has an infinite long sharpened tail called cusp. But we want to arrive at K somehow or other. So I asked Jeff Weeks for help. First, cut off the tail at far away where radius is very small and almost constant. Cap the cut end with a hemisphere.
Yellow one is the hemisphere.
And The area of it is 2 . Tangent at the skirt of Pseudsphere is perpendicular to its axis. So you may suppose Red Pseudsphere with Yellow cap smoothly came out from a wall left. And the total curvature of capped Pseudsphere is zero. Now let the curvature of Red Pseudsphere be , and the area of it be .