As much as I hate to say it, the king would likely win. You have to consider things like experience, training, power, and wit - all of which the Emperor has in droves. However, the King is stupidly strong with his magic, and that can easily override the advantages the Emperor has over the King. But, Sidious is by far the most powerful (if not, among the most powerful) force-sensitives and lightsaber masters in Star Wars canon. Combined with a sharp cunning (he manipulated his sworn enemies by posing as their leader - before killing off the government and Jedi Order and taking over) and you have a very dangerous opponent that could very well defeat the King by a battle of wits rather than a clash of blades. Sidious' Force powers are also notable, and his (arguably) most powerful weapon is the force lightning he can summon from his fingertips. Sidious' own patented brand of lightning was strong enough to make its victims bones show through the skin and muscle and fat, and is practically undodgeable at close range - with lightning being shockingly fast, not to mention the lightning spreading outwards into smaller forks as well. Galen Marek was a particularly notable case of force lightning - his lightning could kill someone on contact, and he was still inferior to Sidious.
Now we have lightsabers. If what you said is true, and Sidious' dual lightsabers do not cut through the King's sword, then Sidious will have a bit of trouble. However, lightsabers are made of plasma, which is actually really damn hot. A simple sword will only cut through flesh, and Sidious can probably still fight with that cut. A lightsaber, however, will kill someone with a single hit. If you get stabbed with a lightsaber, your organs will sublimate, incinerate, and be burned - which means that if you're still alive after the first blow, you're going to die in slow agony.
Now, again, why do I say the King has a better chance of winning?
Nothing protects you from magical incineration where you stand.
