Hi, Sue!
I think your family might enjoy a
Family Suite at Disney's All-Star Music. These budget-friendly suites have a separate bedroom with a queen bed (sorry, not a king!) and a living room with pull out beds for your twins. The Family Suites also have 2 full bathrooms and a kitchenette with sink, small refrigerator, microwave, and coffee maker.
The Family Suites are also available at Disney's Art of Animation, but these are a bit more costly.
If you definitely want that King Size bed, you might want to look into booking 2 connecting rooms, with one room having a King Size bed. Your least expensive option here will be to book at a Value Resort. There are not a huge amount of King rooms, so book as soon as possible if you are going to go this route.
Let me know if this sounds like it might work for you! If not, I can come up with some other options. With so many choices, I hope you find the perfect room for your magical vacation!