This page details two different sorts of top-flush toilet repairs:
Unscrew and remove the knob from the end of the rod that goes through the top of the tank cover. Spray lubricant on the rubber grommet that seals the rod where it goes into the hole in the cover. 2
Push down on the end of the rod as you lift up the cover. Take the cover off the tank. If necessary, lift up the cover enough to grip the rod at the underside of the cover, using vice grips. Lift up on the cover and pull the rod out of the grommet. Set the tank cover aside. 3
Fit the vice grips onto the rod and rotate the rod a bit each way several times as you pull upward. In some cases, mineral deposits can build up and prevent the rod from pulling upward. If the toilet flushes, reattach the tank cover in the reverse order and screw on the knob. Alternatively, if the toilet doesn't flush, enlist a plumber to install a new flush valve and rod mechanism.
The quote above is for the "rod-activated" version.
You mentioned that the knob feels tough-- try using a penetrating oil on it, it may help. It appears as if the rod sits inside of that knob... It also seems to imply that you'd have to pull it for it to work (yours doesn't look like it would be the button type).