At this point, the faucet is likely to be unsalvageable. Your "quarter turn" was not the probably not aerator turning in its threads, but the soft metal twisting before breaking. Especially if the water was "shooting out" from holes that were corroded in the metal. An alternate, equally bleak possibility is that the aerator was cross-threaded.
OK, now that I've level-set, the first thing that you want to do is get actual penetrating oil. WD-40 is a lubricant and water dispersant, but it won't do anything to dissolve corrosion. Buy the type that comes in a spray can, because it will be much easier to apply.
Spray the penetrating oil inside the faucet around the aerator, and let it sit overnight. You want to completely douse the area, so that you have the best chances of oil getting into the threads (and yes, it will take some time to flush it all away once you get the faucet working again).
Set up a mirror in the sink so that you can see what you're doing.
Using a small cold chisel and hammer, tap the remaining piece of the aerator. You want to apply tangential force, in the direction of loosening (and offhand, I can't remember if it's right- or left-hand threaded). You don't have to hit that hard; the goal is to break the threads loose. However, you'll probably have to keep tapping until the broken piece comes completely free.
If you have a place to set the chisel (such as a jagged edge from tearing the metal). that will help. If not, you can create such an edge with a Dremel tool and cutoff wheel. Just be careful not to cut into the threads in the faucet body.