It is possible that she has Coccidiosis. This disease often strikes when chicks are four to eight weeks old. It causes sleepiness, paleness, bloody droppings, lack of appetite, and droopiness. Coccidiosis can kill quickly, so I'd get a coccidiostat as soon as possible. Corid 9.6% liquid (sold for cattle) works best, as it kills all nine strains of Coccidiosis. The dosage for this Corid is 2 tsps in one gallon of water for five days.
During the Corid, you can also try giving your chick electrolytes. However, don't give her vitamins, as the protozoa that cause Coccidiosis feed on vitamins. In fact, the way that Corid works is that it inhibits Riboflavin.
Continue to give your chick sugarwater until you get Corid. After that, only give her the Corid water to make sure that she gets enough of it. Keep her energy up by trying to give her scrambled eggs, as eggs are very good for chicks.
It is possible that she just has something wrong internally that is only showing itself fully now. In this case, there is nothing you can do, unfortunately.
Good luck with your little chick! I hope she gets better! Please keep us updated.