She's laying eggs because her body is producing them. Her body is producing them because it wants her to have offspring and perceives the conditions as favorable.
You can eat the eggs, you can throw them out or you can leave them in the cage for a couple of weeks at a time to see if she'll go broody. You can also boil them, crush them along with the shell and feed them back to her - they are very nutritious and she'll get calcium from the shell.
You will only need to change her feed, if her current feed isn't good enough. She needs something with 20+ % protein, vitamins and minerals and you'll probably need to supply her with extra calcium. I use powdered oyster shell and place it in a separate bowl so the male doesn't have to eat it - it'll only stress his organs unnecessarily.
Some buttons (another name for King quail) just randomly lay their eggs somewhere in the cage. Mine seem to do this with their first eggs, but then they decide on a spot they like and lay most of their eggs there. Once they are laying their eggs in the came place, in my experience there is a fair chance they'll go broody if you leave the eggs there. Quail in general seem to be very sensitive to their surroundings when it comes to broodiness, some never go broody, some will incubate their eggs for a few days and abandon them, some will get chicks and then attack them. And some are excellent parents. In my experience, there is a good chance of success if they have lots of room and some kind of cover in the corners of the cage (I use spruce branches). Mine tend to lay their eggs under those spruce branches but they usually sleep in an open area, until they go broody - then the broody hen will start sleeping and spending most of the day on the nest.
You won't 'know what her plan is' until she goes broody - if she does. But as mentioned, if she keeps laying them in the same spot, in my experience that's a good sign she might go broody. I'd leave the eggs there till she has 10-12 eggs in the nest. If she hasn't gone broody by then, there are likely too many eggs for her to cover and the fertility (I'm assuming there is a male as well?) of the oldest eggs will start dropping, so I'll just remove the eggs and let her start over if she pleases.
With regards to cleaning the cage, that all depends on the size of the cage, position of the nest, type of bedding and personality of the hen. I can't really tell you she won't mind at all if you clean the cage. She might. Or she might not. If the cage is large and the nest is placed far from the door, perhaps she won't mind if you just leave the nest alone and clean the rest of the cage. But my cages can go quite a while without being cleaned, so if you want the best chances of getting chicks, I would just leave the cage as it is and see if she goes broody.
Otherwise, powdered egg shells from chicken eggs are fine too.
