You're doing just fine. Just change the dressing once a day or so, or if needed. Be sure the dressing does not put pressure on the wound; make it just a little loose there. The Neosporin is fine.
Betadine (please dilute) and peroxide both kill some new tissue growth, but either is great for initial cleaning of a wound because they also kill germs. I would not even clean the wound after the initial cleaning unless you see redness or pus; then a little peroxide or dilute Betadine is fine, just pour some water afterwards to rinse it away. (For a large wound, I would say saline instead of water, but don't feel that is needed here.) (Old worn out nurse talking here.)
Personally, I would not give a systemic antibiotic for this, simply because chickens heal amazingly well. They are often recommended after major wounds, so it is your choice. Most likely she will do fine without this. If you decide not to, but you then start seeing pus or redness on or near the wound, that would definitely be the time for antibiotics.
And please don't beat yourself up over this. I promise you that many of us could say, oh yeah, this is what I did to my chicken/kitten/whatever. Including me! Accidents happen.
Feel free to PM if needed.