Painful Splitting Of Toenail In Dog,can You Help?

She has some sort of dermatological condition that makes her nails either too soft or too brittle, so they split.  If you find a split before it reaches the quick I would seal it up carefully with superglue (read the instructions!).

If it has reached the quick you need to take antiseptic precautions and protect the foot while the nail heals, possibly with bootees. Try supplementing her diet with sources of collagen (e.g. Gelatine) and zinc, which is supposed to be good for strengthening nails.

Bootees are easy to make: Choose a strong material like canvas, and draw 2 C shapes that are larger than her foot all round by about an inch, and have a long enough "neck" to reach her wrist/hock.  Stitch them together so you have a "sock" when you turn it inside out.  You can either sew velcro to it as a fastening, or just tape it on.  Do not use elastic as this can damage the circulation of blood.

I would not use super glue on your dog's nails.  
I am concerned that your dog has an underlying problem leading to the fragile, brittle nails.  If the nails are not too long and multiple feet/nails are affected this raises suspicion that there is an underlying problem.  Have your dog rechecked by your veterinarian.  They may decide to biopsy the nail bed for culture (bacterial and fungal) and histopathology (examining stained sections under the microscope).