Hello Lily's mom. I'm Dr. Deb and will do my best to help you today.
I'm sorry that Lily was so energetic that she hurt her dew claws although this is a fairly common problem as you might expect.
The following is what I'd suggest although you'll want someone helping you since this may be a tad uncomfortable for her.
1. For the split nail, I'd just trim it a little but that's about it. As the nail grows out, continue to trim it so she doesn't snag it on anything.
2. As for the nail which is folded back, I'd also cut off as much you can up to the "fold" if possible. This might reveal some of the tissue or "quick" underneath but that's ok.
3. You can gently clean the folded nail twice a day with over the counter dilute Betadine or Iodine (to the color of weak tea).
I'd avoid peroxide which can retard healing; even warm water will be preferable.
4. You could even soak the toe in a warm Epsom Salt bath but this isn't usually necessary in my experience.
5. If she's not too uncomfortable, you can apply a small amount of over the counter antibiotic ointment if there's a raw area exposed.
6. An even better product for this problem would be Vetericyn Spray which can improve healing time. It's often available at most local pet or grain stores or online.
Since it's a spray, it will be easy to apply; it's also non-toxic so won't be harmful if ingested.
7. It's going to be important to prevent Lily from licking the toe/nail since she can make it worse. Either she'll need to wear a cone although I know she won't thank me for suggesting it or you can put a sock on her foot or a light bandage.
I don't usually advise the latter since it's often difficult to get them to stay on without making them too tight. Then circulation is compromised and the foot swells up.
8. As long as she's not vomiting nor taking any steroid or non-steroidalanti-inflammatory medication, Aspirin can be given at a dose of 10 mgs pe rpound of body weight twice a day with food to avoid stomach upset.
In most cases, these injuries will heal just fine and most patients rarely even limp or pay much attention to the toenail after a day or so. The nail will regrow although it may take several weeks to do so.
If the area should become swollen or there's a discharge, then she may need antibiotics but this is fairly rare in my experience.
I hope this helps. Deb