My husband sort of did battle with it until it opened and he could fix the problem. Over the next while whenever I was in that bathroom I fiddled with it until eventually I had it reassembled. In the process, I discovered a magical truth. Those buttons screw off.
Just pinch the outside and turn anti-clockwise (lefty loosey). At first it will seem like it's not going to do anything. But keep turning and eventually.
At this point you can just lift the top of the cistern like an old fashioned flush on the front model and get at the whatever you were trying to get at inside.
A while later the other toilet in the house started flushing constantly. The cistern was constantly filling up and emptying at the same time, wasting huge amounts of water. So I simply unscrewed the buttons, lifted off the top and had a look. That black thing at the right had drifted up, and water was flowing over the top of the white pipe in the middle. So I pushed the black thing down, put the lid back on, screwed the buttons back on, and felt very satisfied with myself.
Fixed the toilet. Like a boss.
— Saoilí (@saoili) August 13, 2013