First of all, how old is your niece? I am assuming she is not that mature enough to understand your dad's current situation. You can actually get one but strictly stay it away from your dad. You can build a customized dog house and leave it in your garage or garden. In this way, you can satisfy the happiness of your niece without bringing risk to your dad's recovery. Just make sure to remind and stay an eye on your niece not to bring it inside the house. There are lots of dog-friendly areas in the metro where you could drive your niece to play with her new found friend.
I guess you can buy her a substitute instead? A stuffed toy dog, maybe? Or anything that can keep her distracted from wanting an actual pet dog. You can discuss to her as well that she can get a pet when your dad has finally recovered.