This quote nicely sums up the situation:
“This is what happens when an unstoppable force meets an immovable object. You truly are incorruptible, aren't you? You won't kill me out of some misplaced sense of self-righteousness. And I won't kill you because you're just too much fun. I think you and I are destined to do this forever.”
The author of the article you linked to then makes a sound argument for why the Joker should be killed:
Gothem city would be much safer if the Joker were killed.
The Joker is a clear example of “an unstoppable” force that can be seen to harm. The Joker proves time and again his disregard for law, any semblance of respect for other lives, and his consistent need to create chaos in a methodical way. No prison can hold him, no punishment will effect him, no treatment will cure him. All have been tried, all quiver into dust or, like Harley Quinn, are transformed into another tool for his plans.
A guaranteed way to prevent any more of these horrible crimes is to end the life that creates them. This is a choice to kill, to severe forever the final thread of life, to close the final door to any problems caused by and to the entity in question;
Should the Joker be put to death? Certainly. But should Batman do it? No.
Batman would never kill even a supervillain like the Joker, unless it was in self defense. To suggest that he'd not only kill but then cover up the deed is completely contrary to the nature of Batman and would forever change who he is.
Were Batman to give in to his dark side, even to dispense justice towards someone who clearly deserves it like the Joker, he would no longer be the uncorruptable guardian. Indeed, the Joker would likely die happy if Batman killed him knowing that he'd forced Batman to take the first step down the road towards the chaos that he loves and that the next step would be that much easier.
Batman will capture the Joker and turn him over to the proper authorities. It's then up to the state to try him before a jury of his peers and execute him once he's convicted.
The problem for the author is that he opposes the death penalty. Rather than re-evaluate his principles to make room for even extreme cases, he chooses to corrupt the uncorruptable.