Sadly it is instinct to chase anything small that runs, they even chase a plastic carrier bag as a lure at some of the tracks. There is no danger whatsover to a child, other than to a tiny crying baby which should not be left alone with any dog.
Make sure the fence is high enough so it cannot jump over and tell your neighbour to muzzle the dog when out, as they can easily grab a cat hiding under a hedge or a car, if she is not vigilant, andif she is elderly she may not be strong enough to hold it. They do make fabulous pets, provided you take the necessary precautions. I have rehomed greyhounds with people with cats but the cats are used to dogs and don't run away.
It is a very sad story, but unfortunately one of those things that has happened and cannot be altered.
I have small dogs, and if a cat got into my garden I am afraid they would do the same to it. My local cats stalk along the fence taking the p*ss out of them.