Preheat oven to 200 degrees F, and set a timer for 10 minutes.
When the 10-minute timer goes off, turn off the oven, leaving the door closed.
Next, pour the quick rise instant dry yeast into a bowl with the warm water. Stir to incorporate, and set aside.
Heat the milk in a small pot or microwave until just heated through.
Remove from heat, and add the vinegar and lemon juice.
Pour into the milk/vinegar/lemon juice bowl, and the yeast/water bowl into a Kitchenaid stand mixer fitted with the paddle attachment. Add the eggs, sour cream, butter, cinnamon, nutmeg, vanilla extract and sugar to the bowl.
Then, add the rice flour blend, one half cup at a time, to the bowl.
Process over medium speed for 1 minute, then on high speed for one minute.
Remove the paddle attachment from the bowl, and replace it with the dough hook.
Knead on Kitchenaid speed setting 2 for 8 minutes. (This is the most important step of the process. You are kneading the dough for texture, so it will taste just like a traditional King cake. But, you have to knead it exactly at setting 2 or the texture will be off.)
After kneading, remove the bowl from the stand mixer. Place plastic wrap loosely over the bowl.
Make sure that the heating element in your oven is not red anymore. The oven should still be warm but not hot.
Place the covered stand mixer bowl in the oven.
Let rest for 1 ½ hours.
Remove the bowl, and pour the batter into a large glass measuring cup.
Preheat oven to 375 degrees F.
You should have about 4 cups of batter.
Clean the stand mixer bowl, and add all of the filling ingredients into the bowl, except for the blueberry preserves.
Process on medium speed for about 1 minute, or until all of the ingredients combine, and the cream cheese is slightly whipped.
Grease, and use more of the gluten free rice flour blend to flour a bundt cake pan. (Be very liberal with the oil or butter and flour, so that the cake will slide right out of the pan after it cools.)
Pour half of the batter into the bottom of the pan. Pour the cream cheese mixture on top of the batter.
Next, pour the blueberry preserves on top of the cream cheese mixture. Keep the filling as close to the middle of the cake as possible throughout the process.
Pour the rest of the batter on top of the filling, making sure that the batter completely covers the filling.
Bake for 30 minutes, or until cooked through. You can use a knife or cake tester to make sure that the very edge of the cake, or the very inside of the cake around the bundt cake hole, is cooked all the way through.
Cool in cake pan for 30 minutes to 1 hour, then place a large cooling rack over the exposed cake. Very, very carefully flip the cake over so that the bottom is resting on the cooling rack.
Make the icing by whisking the butter, 2 tablespoons of milk, and 2 cups of confectioner's sugar together in a large bowl. This will make a thin icing. If you want a thicker icing to use less colored sugar sprinkles, then add the other 2 cups of confectioner's sugar to the mixture. Pour 2 tablespoons of the hot water into the mixture, adding more as necessary, to achieve the consistency you desire.
Pour the icing in circles around the top of the cake. Shake the plate slightly to encourage the icing to fall down the sides of the cake.
Finish with the purple, green, and gold colored sugar sprinkles. To make the sprinkles yourself, add about 3 drops of yellow food coloring to ½ cup sugar, then stir to combine. Do the same with the green food coloring for the green sprinkles. Make the purple sprinkles by combining 2 to 3 drops red food coloring with 1 to 2 drops blue food coloring. Stir well, then use a spoon to incorporate into the sugar.
Serve or refrigerate the gluten free king cake immediately.