Well me and my husband are lovers of piercings. Earring, face, body the whole nine yards. My husband has gotten his lip infected and my eyebrow got infected. Both were becuase we got lazy and didn't clean them properly for a few days when they were newer. Not a bad record for how many we both have.
First STOP moving the piercing around. That is bad! Don't use hydrogen peroxide. I know it makes logical sense but it has been shown to make matters worse. Also DO NOT remove the piercing. It acts now as a shunt that allows the infection to come out. If you take the earring out it will close up and guess what you'll trap that lovely infection inside her little ear. So if you decide to take it out do so AFTER you cleared up the infection.
Now the best thing to use I've found is Sea Salt. It clears up the infection pretty fast if you do it deligently. First keep washing with an anti bacterial soap about 4 times a day. Nothing else! Then after you wash with the anti bacterial soap take about 16 oz of water and dissolve in about 2 tablespoons of sea salt. Put it on a q tip and put it all over the affected area.
You should start seeing results rather quick.
If you want more reassurance contact a professional piercer. Most tattoo shops have one on duty. Also, don't ever bring your children to the mall to use a gun for their little ears. It is so unsanitary. A real piercer doesn't use a gun because they can't be sterilized in a proper way. You can't put plastic in an autoclave... nor do mall stores have one! There can be small amounts of blood on them from every piercing they do. They think if they use new earrings on each person and wipe them off with an antibacterial wash they are safe. Not my idea of sanitary. Plus these are 19 year old girls half the time who were trained for about 5 mins on how to use the gun then let loose on little girls. Bad formula by any logic. But that is neiter here nor there. Just something to think about if you have other children or you decide to get her ears pierced twice.