If you can find the burn pads that Sara B mentioned, I'd get some. A couple of years ago I did something really dumb and burned my finger tips really badly. I had made a breakfast casserole in a handled skillet (in the oven) and taken it out and set it on the counter. Then I turned around and did something else, then turned back and picked up the pan, one hand on the long handle, one hand on the little short handled side. No potholders. (you see, I RARELY use this pan in the oven. Usually it is stove top only, so the handles wouldn't typically be hot. And I had already taken it out of the oven and set the potholder down, and then wasn't thinking).
Anyway... I let go with one hand really quickly, but it is a heavy bottomed metal skillet, full of eggs so it was even heavier, and I have tile floors. If I had let it drop it would have cracked my floor. So I more or less tossed it onto the countertop... but I still held onto it longer than I should have with that one hand. My finger tips were burnt pretty badly. I didn't realize at the time how much, because all burns hurt. But usually, after 10 minutes the burn doesn't hurt so much if you've run it under cool water and applied aloe or whatever.
Well, (we had guests and I had sent them on ahead to the reunion without me) 30 minutes later trying to get a shower, it still was terribly painful and could NOT use my hand/finger tips. If I took it out of the water it burned. Even 30 minutes later.
I ended up filling a plastic cup with ice water, wrapping it with a damp paper towel, and holding the cup with that hand (to keep the finger tips moist and cool via the paper towel) while I drove myself to the nearest CVS. They had those burn gel pads and I bought them. I had to wait until I got to the reunion for my SIL to apply them for me, though, because I couldn't do it myself with my burned fingers. As soon as she placed one on my fingertip, it was INSTANTLY 90% pain relief. I couldn't wait for her to finish the other fingers! They ALL had the same relief. It was wonderful. (And I didn't end up with any blisters either. They were completely healed in about a week. The outside layer of skin peeled on a couple of the finger tips, but that was all. And no scars.
I have never found those pads in a store or pharmacy again since. And I have looked, because I want to always keep some on hand at home now. But I can't find them.
They WORK.
But, you really should take your daughter to be seen by the doctor. I have read that anytime a child is burned worse than 1st degree on areas like hands/feet/ joints that they should be seen, because of the possibility of scarring interfering with growth.
ETA: I read the same email as Karen about the egg whites and I started to mention it, but I hadn't tried it out, so @Karen, thanks for the first hand info!! I'll try that next time!