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The first of the major two being that you need to have a net war success of at least 40, this is very easy with nukes as each gives a flat +10.
Second is that your enemy needs to have a power rating below the world's average. This is the one that takes more effort to clear, and the one that causes questions to be asked as it means that the one city civ you may have capitulated previously can effectively block future capitulations.
so if you already have the territory and population destroying his military with nukes can enable capitulation.
generally though I just do a naval assault, razing 4 coastal cities with a conventional army reduces his territory and population and causes rapid capitulation. you could I suppose soften the targets with nukes but its overkill.
It seems most likely that their power is well above the world average threshold (not sure if its world average itself or 2/3rds of it). If this is the case the excuse given will be "We're doing fine on our own".
The battered rumps of capitulated civs or the unusual small colony are notorious for dragging world power down so much that you can practically need to annihilate civs to force capitulations later. Them having a vassal themselves will inflate their perceived power making it worse. Sounds like your best bet is to invade.
a Vassal adds to the master's power and territory. to break a vassal from its master you need to either kill the vassal's territory to 50% of what it was at the time of vassaling, or make it larger than the master by a substantial amount.
it wonks the numbers. I've definately capitulated a master, and then the vassal leaves, and then I capitulate the vassal if I've engaged its army. on the same turn.
its just more messy.