Welcome to the forums!
I'm sorry you are having problems with your budgie. Given the fact she got ill shortly after laying the eggs it just proves she was clearly not in good shape in terms of health to undergo through mating/breeding/raising a clutch.
In order to have a successful breeding, a budgie couple needs to be in top physical health and in breeding condition, they must also be under a special breeding diet with plenty of calcium.
Since the eggs have not been incubated, there is no life formed inside them and my advice is to remove the eggs as well as the nest box and give your female budgie more time to recuperate, she may seem well now, but breeding takes a lot out of budgie, especially the females who do most of the work: laying eggs, incubation and feeding during the first weeks of a chick's life. And letting her sit on eggs that aren't fertile or letting her go for another round of eggs is not wise at this point, because she may very well have a relapse and fall ill again.
Give her a few months rest, continue to give her a good diet and try again later, if that is your wish.
For more detailed information, check the stickies on the Breeding section of the forums and also on the Diet and Nutrition section.
Good luck with everything!