I kidnap your doctor’s wife and torture her mercilessly while forcing him to watch, until he can be persuaded to introduce a subtle mind-control substance into your system. You then carry on as before, but now I am the unknown power behind the throne and you do solely as I command.
As reward for his cooperation, I reunite your doctor with his wife, who is revealed to be alive and unharmed. At least, that’s what the subtle mind-control substance encourages him to think. Can’t have him loose-cannoning around possibly to be used against me.
In due course I extract the location of the nuclear device from your dribblingly incoherent mind, and the two of you are both removed from their current location to a delightful South Seas atoll where we celebrate the invention of the skimpy two-piece bathing suit in a poetically satisfying fashion, if you get my drift.
I enjoy a comfortable retirement in complete anonymity, with only a vast personal fortune and a staff of gorgeous and highly loyal retainers to lighten my declining years.
Magical, no, but the nation’s bureacracy is centered there. Remember, in the OP I said the God-King left the infrastructure intact. Apart from the fact that the GK’s holding half a million people hostage to his own shittiness, the destruction of the nation’s capital wouldn’t be anythng other than devastating. Especially since you KNOW that nuke is going to be as dirty as possible. Anybody who’d do such a thing is going to want as much collateral environmental damage as possible.
Sorry; you have no access to super-tech or magic, remember? (Unless you steal it from the God-king, but since he’s not using it to enforce his will I think we can assume he doesn’t have it either.) Unless you can demonstrate that this suble mind-control substance of yours already exists, it’s disallowed.
But if you can demonstrate that, I think you have a good plan, though Revenant Threshold has the best plan by far.
Oakminster, the problem with your plan and the other single-conspirator assassination plots is that you’re wasting an invaluable resource. I don’t mean the capital, either: I mean YOURSELF. As the situation is set up, the God-King is planning on his regime collapsing on his death anyway. Simply assassinating him, solo, causes the collapse of the government when no one is in a position to take the best advantage of it. All that gets you is anarchy.
Answering my own question: I’m with whoever wanted to take The Moon is a Harsh Mistress as as model. For the most part I’d use a plan like Revenant’s, except that I’d be setting up a “B” plan as well. Specifically, I’d be assuming that my insurrection is crushed at some point,and I’d be grooming someone to take the reins when the God-King dies and the capital is obliterated. And, as I’ve said before, the very first thing to do is to recruit the GK’s personal physician to my cause.
Since Australia’s capital is Canberra- and everyone hates Canberra- I’d simply cut off the power and water to the city, and wait until Skald The Evil died of starvation, removing Canberra from the face of the map and allowing me to step in as His Imperial Majesty Emperor Martini I, Supreme Overlord Of The Australian Empire, and begin my own plan of Neo-Colonialism on a scale that will be both awe-inspiring and terrible to behold in the eyes of those who would dare oppose me. 
Come on guys, he said he was a GODKING so you’ve got to analogize his control over the country at the Fidel Castro/Kim Il Sung level. He has LEGIONS OF GENETICALLY-ENGINEERED FLAME-BREATHING VENOM-SPEWING FLESHEATING WINGED HOWLER MONKEYS flying around enforcing his will. There isn’t going to be any popular uprising, you aren’t going to have easy access to explosives and sniper rifles. His CIA, FBI and NSA can wiretap every conversation in the country. It will be almost impossible to set up an effective conspiracy.
Is the point to merely get rid of the godking or somehow deal with creating a new regime and ridding our country of fire-breathing flying monkeys.
If you didn’t care about the nuclear bomb then you could probably smuggle a sniper rifle and kill him from far away, who knows you might get lucky and leave a living yet brain-dead body there (you might want to let other people know about this dead man’s switch at this point so they don’t pull the plug).
If you want to avoid killing everyone in DC then don’t you need to capture the Godking alive? How the heck do you do that under a suppressive regime that is supported by flying fire breathing monkeys? You need to get close to him, real close. You need to control some aspect of his life without his knowing it. Can you subvert his doctors? Can you subvert his bodyguards? Can you subvert his family? All you need is one of these and then the answer becomes simple. If everyone around him is fiercely loyal then you have one chance to create enough chaos (on a level they never planned for) for their security protocol to break down (you won’t get a second chance), you need something that will make everyone run like mad without thinking about their duty (at least temporarily); I would release a (well publicized) painful penis and testicle eating virus in the White House ventilation system (hopefully this regime is backwards enough not to hire anyone other than straight males who place the majority of their self identity in their bits and peices) and use female commandos (some of the despoiled virgins will work nicely) in the ensuing chaos to innocuously drug and extract the GodKing.
Someone mentioned that the kill switch could either be a constant (or intermittent transmission) that would set off the bomb if the transmission were interrupted. This is unlikle considering the fact that this despot along with every other despot in history has a EMP proof bunker that they visit from time to time whenever they think something bad might happen. So you know all you have to do is block the outgoing transmission. I would probably give those female commandos a lead lined hood (or something that would block a transmission) in case they reached and captured the GodKing but couldn’t extract him (slip on the hood and slit his throat). If we successfully extract the GodKing, we would then take him to a deep EMP-proof bunker and execute him and release the video on the internet.
Of course this just means that someone else might step into his place but who would the flying monkey listen to now? Perhaps Canada and Mexico (sensing an advantage) would invade and rid us of our Flying Monkeys. I just don’t know enough about the flying monkeys, how they are handled, how they are controlled, are they sentient, are they autonomous, can they be negotiated with, would they eat poisoned bananas left lying around on people’s rooftops?