To start off, note that I will be pretending that the Argent Dawn and Argent Crusade are the same. They aren't, but close enough.
At first glance this organization will soon be dead. Not defeated, but instead victorious, and lost. With Arthas gone there would likely be little left to do in their specialty: fighting the Scourge. They would become like Maiev after killing Illidan, what is she without her hunt?
This will not happen. The Argent Dawn will not be finished.
For starters, we don't know what happens when the Lich King is destroyed. The best case scenario would result in something like the Forsaken, a mass emancipation of the undead, who then turn on their living overlords and destroy them. This isn't likely. The Scourge is not purely directed by the direct mental control from the Lich King. While there is likely to be a mass breakaway, I don't see the majority even having the sentience to fight back. Instead many are mindless and will simply be re-enslaved by the nearest necromancer.
I foresee a civil war within the Scourge. The 'nicest' faction will be the sentient undead who fight for revenge; a sort of second Forsaken. Then there will be the selfish types, those who joined for power. Finally there will be the zealots; these would include many of the early peasant recruits who joined the Cult of the Damned in direct opposition to the Light and life. None of these groups are likely to be kind to the living, or the other way around. The Argent Dawn will have plenty of work left to do in ending the civil war and moping up the remnants of the Scourge.
The Argent Dawn is not limited to fighting the Scourge. I don't know if they know that the Scourge is from demons, but they certainly know of the corruption and dander of demons. They know that the Scarlet Crusade was infiltrated and run by demons. More directly, they were at the opening of the Dark Portal, fighting off the waves of demons. There is still plenty of demonic corruption for them to fight in Azeroth. If they get bored enough, they can follow the Cenarion Expedition to Outland.
There are other threats waiting as well. Do we honestly think that Deathwing's return will not provide another cause for the Dawn? I don't think for a second that they'll see the Eastern Kingdoms split in two, shrug, and say "Well, he'd not undead, so I guess we're done."
Tirion might be coming to an end, but the Argent Dawn is not.
2 hours ago