I dunno, "discontent" is a pretty politically charged word, going back to Shakespeare - "Now is the winter of our discontent" from Richard III is referring to an attempted political overthrow.
It's referring to a successful political overthrow.
The quote really needs the first two lines:
Now is the winter of our discontent
made glorious summer by the sun of York.
The verb in the sentence is "is made", not just "is". "Now" it is summer, not winter. They were discontent in the past. Now they are happy.
York (Richard's brother, Edward, now King Edward IV) has overthrown King Henry VI. There's also an important pun: "York" also refers to their father, also named Richard, who was the Duke of York until his death at the hands of Henry's faction. So Edward is also the "son of York".
That said, Richard is being sarcastic. He's plotting the next political overthrow, which will also be successful. And who will in turn be overthrown again. That, at least, will put an end to it, if for no other reason than that literally everybody else is dead.
Unhappiness sounds much more pedestrian.