How do I open up a B&Q Rejuvenate Close-Coupled Cistern?

I can unscrew the top, which is around the push-button, but then it only comes up so far before it stops because the button is held in place with the bit it's connected to inside.

I want to open it up to find out why it's taking a long time to fill (I've looked up some of the possible answers on here) but can't actually get access to it. Any help is much appreciated.