The same thing happened to my three year old in June. It's just mosquitos believe it or not.
Forgive the long story, but I think it will explain a lot. We always visit my in-laws in June who live in a wooded area. When we got home, since it was the summer, my daughter wore light sleeveless pajamas to bed. She woke up the next day with the huge bullseye spots on her thigh and upper arm. At first I thought it could be ringworm, but it didn't match all the symptoms.
The next day, wearing the same pajamas, she had another huge bullseye on her other arm and another one on her leg. That was Sunday. Now I thought for sure it was lime disease because of where my in-laws live, so I decided to take her to the doctor first thing Monday morning.
Sunday night my daughter wore her long sleeve pajamas with pants to bed and woke up with two huge bullseye spots on the top of each hand.
When we got to the doctor, the doctor said it was just mosquito bites. I couldn't believe it after the strange progression of bullseye spots and them getting bigger with each passing day. But my doctor said that there was probably one or two mosquitos that got into her room and that come out at night to feed on her (yuck). She said they can live indoors for a long time as long as they have something to feed on. That's why the spots popped up on her hands the last day, because she was wearing the long sleeves and her hands were the only skin that was exposed.
As for the huge bullseye spots, she said some kids are just more allergic to the bite than others. That's why some will only get a red dot, others get the big bullseye.
After that, my daughter slept with my husband and I for the next few nights just to be safe, and after that, no more bites, and the bullseyes went away pretty quick.
I'm sorry your daughter's bites have gotten infected. That doesn't sound very fun. If they are still itching her, you can make a paste of baking soda and water and apply it to the bites. That should help stop her from wanting to scratch them. But hopefully this will give you a little peace of mind that it's nothing more serious. Feel free to PM me if you have any questions. Good luck!