Hello, this is Dr Weaver I am an experienced equine vet and I hope I can help your mare out. I am sorry to hear about her injuries, it sounds like she had a pretty bad accident. You say the injuries happened about a week ago correct? Have you had her evaluated by a veterinarian?
A couple questions for you, is the mare walking comfortably on the leg that was cut or is she limping? Are any of the lacerations near a joint? If you can maybe you can send me a picture.
As far as inability to lift her head, I would be concerned about a neck injury, horses can subluxate their neck vertebrae, or fracture portions of the vertebrae. Does she wobble at all when she walks? Strong anti-inflammatory medications may be of benefit at this point, if she has compression of her spinal cord or nerve roots. You would need to get those prescribed by a veterinarian.
How is her eye looking? If she is squinting or watering, it is best to get it looked at by a vet to stain the eye and get her on appropriate eye medications.
If your daughter thinks the mare needs antibiotics, penicillin may not be the most appropriate antibiotic. I would encourage you to get her evaluated by a veterinarian.
That being said if you do decide to give the penicillin, make sure your daughter is careful with it, some horses can have a reaction to it that looks like they are having a seizure (it's called a procaine reaction). You never want to give penicillin in a blood vessel, it's always important to draw back on the syringe and make sure you don't see blood coming back before you inject it into the muscle. If you do see blood redirect you needle to a different site. Always give penicillin while your horse is in a well bedded stall or on soft ground in case they have a reaction and go down. The dose for penicillin is 3cc per 100 lbs, so a 1000lb horse would get 30cc, twice a day, IM.
-Dr Emily Weaver