This is a good question; but it is made difficult by the unfeasibly high solubility of potassium nitrate in water. The quickest way might be a density measurement.
As you know, in a supersaturated solution, the solution contains MORE solute than that which would be in equilibrium with undissolved solute. Scratching the sides of the sides of the container with a glass rod, should precipitate out masses of crystalline solute from a SUPERSATURATED solution.
For a SATURATED solution, a density measurement might be the most appropriate indicator. So here, ascertain the mass of an accurately measured volume of the salt solution. If the measured density is over
So why is potassium nitrate so soluble? In fact most (all?) nitrates are soluble because of the charge density of the anion. The negative charge is distributed over 4 centres, and thus dissolution is entropically favoured over the dissolution of anions (cf halides) where the negative charge is localized.
As an interesting fact, you can collect very pure crystals of saltpetre, potassium nitrate, if you have access to horse or cow manure. Turn over the older clods, and you usually find beautiful macroscopic crystals of potassium nitrate. Urine, even human urine, is another source, but that's a little bit more offensive than cow or horse poo.