Post-war, the Earth Kingdom is usually depicted as peaceful, but history suggests many ways things could go wrong for them, badly wrong.
The Dai Li and Ba Sing Se's civil service are both compromised. The Dai Li actually supported Azula, and there's no telling how many people in the civil service actively supported Long Feng. Both of them will need to be purged by the Earth King, since he can't trust them, but they did also do a useful job.
Without the Dai Li, suppressing any riots will be much harder, as might be collecting taxes. If Long Feng had any sense, the senior people in the treasury will all have been personally loyal to him. Remove them, and sorting out the city's finances becomes a challenge.
If Kuei is competent, all this can be overcome, but he doesn't give that impression. Assuming he remains weak, there will be a power vacuum at the heart of the Earth Kingdom, which is an opportunity for ambitious generals.It wouldn't be difficult for a few of them to convince themselves that they're the right people to 'advise' Kuei.
Considering that the generals are each likely to have a regiment or two behind them (soldiers with more respect for their leader and paymaster than the nominal king) if the generals start quarrelling amongst themselves, or with the civilian leadership, Ba Sing Se is in trouble.
Beyond the city walls, bigger problems lurk.
What happens to the Earth Kingdom's armies? Disbanding them will leave a lot of unemployed young men, trained to fight, seldom a good thing. The worst case is if they are disbanded in place, with no arrangements made to get them home - their pay stopped with a single stroke of the pen. It's folly, but its the kind of thing a second-rate penny-pinching bureaucrat might well do, and all the first rate bureaucrats in Ba Sing Se were working for Long Feng.
If they did that, they get camps of armed men, stranded 100s of miles from home, with no money. No doubt some of them would cause no trouble, but history suggests that others would become brigands, stealing food and money from the peasants, which they'd justify as no less than they deserve for winning the war.
Worse, some regiments might become mercenaries, or the private armies of the generals. It was a common enough pattern after long wars in medieval times - people who've spent their whole life as soldiers may not want to do anything else.
And then there are the fire nation colonies. The obvious thing is to hand them over, but to who? The traditional local rulers may well be dead without heirs, or collaborators. The most likely alternative is to hand them over to BaSing Se, and let the Earth Kingdom appoint a local governor, but even if the city is not struggling with its own internal problems, that would still give it a severe case of administrative overstretch.
In some cases there will be local resistance leaders, who can gain effective control through popular acclaim, but good resistance leaders are often bad rulers.
There are also likely to be major disruptions in trade. The colonies will loose much of the fire nation markets, but be able to sell more freely to teh rest of teh Earth Kingdom. Overall, thsi may be neutral, but their will be economic turmoil before things settle down, never fun.
Overall, a glance at the history of South America and Africa shows how badly decolonisation can go wrong.
The Earth Kingdom wouldn't necessarily suffer from all these problems simultaneously - one or two would be quite enough. The net effect could range from low level disorder - riots in Ba Sing Se, brigandry in the countryside - to open civil war.
This, of course, is only a setting, not a story. The story is how the gang cope with a struggle where there's no clear enemy. The Earth Kingdom's problems can't be fixed by taking out any one man; they're deeper rooted than that.
