So instead of trying to infodump all this into the actual chapters, I figured I would do a rundown here instead of how the state of Westeros at this point in time. Which parts are there? Who rules them? What kind of military forces do they have?
I've used some information from both the books and show to put this together, plus of course some creative license. (please correct me if there are any super obvious errors though)
Below is a list of the kingdoms that remain, who rules them and an approximation of their military power at this point (this is their own standing army, not including bannermen).
The Stormlands: Ruled by our hero and savior Lord Gendry Baratheon. Who now has a decently sized army. He still had the men from the Baratheon armies he rallied in the Neck and in addition to this he also has added the commoners from the capital who chose to fight for him. He is currently completely lacking a fleet though (the Baratheon fleet was wiped out during the wars or incorporated into the Lannister fleet) which could become a problem as the Stormlands have plenty of coastlands.
Approximate army size: 7000 (2000 men from the Baratheon armies + 5000 former commoners), no fleet.
The Reach: Ruled by Lord Paxter Redwyne. He has inherited the strong armies of Highgarden from the Tyrells. However, these forces should be weakened after the battle against the Lannisters. But the Redwynes themselves possess the biggest fleet in the kingdom
Approximate army size: 10 000 men, 1000 ships.
The Westerlands: Ruled by Lord Tyrek Lannister. The Lannister forces are seriously depleted at this point as most of them were taken out as Queen Cersei fell. So Tyrek only has the men he kept at Storm's End and some additional reserves that had been kept in Casterly Rock. But what used to be the strongest and biggest force in the kingdom is now the smallest and weakest.
Approximate army size: 3000 men, no fleet.
The Vale: Ruled by Robin Arryn. Has the biggest forces in the whole kingdom and they are basically untouched by the previous wars. Some of them perished in the war in the North (he sent about 5000 men there) but most of them are still intact and fit to fight. Robin might be the youngest lord (I'd put him at 17 at the start of this story), but he's definitely the strongest.
Approximate army size: 40 000 (yes you read that right! His forces are basically the size of all the others combined), no fleet.
The Riverlands: Ruled by Edmure Tully. Has had his forces depleted in the battles against the Freys. But he has had some time to rebuild since.
Approximate army size: 5000 men, no fleet
Dorne: Ruled by Quentyn Martell. His forces are untouched by the wars in the North. They should also have a decently sized fleet as Dorne is a seaborne nation.
Approximate army size: 20 000 men, 500 ships
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Then there are also the Iron Islands and the Neck, who aren't really big enough to be their own kingdoms but also aren't part of the other kingdoms (since they both used to belong to the North).
The Iron Islands: Ruled by Yara Greyjoy. She still has a good-sized fleet although the men who fought under her uncle haven't rejoined her.
Approximate army size: no army, 500 ships
The Neck: Ruled by Howland Reed. Their forces are not big but they are adapted to fighting in their particular environment of the swamp.
Approximate army size: 1000 men (and a dozen crocodiles), no fleet
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And there's the king who rules the lands from the capital of King's Landing. He of course also needs some forces.
The Kingdom of Westeros: Ruled by Bran Stark. Let's say his sister Sansa left him a good-sized army of her Northern forces to support his reign. But he will have trouble keeping the kingdom under control without a bigger army. The North has no fleet though so he would probably not have any ships at this point.
Approximate army size: 10 000 men, no fleet.