"Activities Director Janie Alexander welcomes Shani Nielson, she said, because she knows they are there to help.  "Shani is very caring with the elderly," she said.  "She's very genuine.  It's not a show. They get a sort of bond with the residents, and they have an ongoing list of the people who really like the visits from the animals."  One of these residents, a man who has been brain-damaged from birth, doesn't respond to people very often.  When Hamlet (cat) made his appearance, however, the man brightened, stroked the cat's fur, and started pointing to one of his books with a cat on the cover.  When the time came to leave, Shani Lynn Nielsen went reluctantly.  "Sometimes it's hard to leave," she said, "when you get a reaction like that.""   Dave Davis   Tribune Staff Writer