At a minimum, you can add insulation to fill the joist spaces in the floor and add resilient channels between the bottom of the floor joists and the drywall. Multiple layers of drywall will help. Though, with a hard surface flooring in your kitchen, you won't likely entirely eliminate noise transfer. You might consider replacing the hardwoods with something more resilient like cork or linoleum. Depending on your budget, it might be worth bringing in someone who specializes in acoustic separation to advise on the best ways to mitigate sound transfer between the two living spaces.

For impact sounds like a dog bone dropping on the floor you would need to decouple the spaces. Right now your flooring is directly connected to the joists which his ceiling is directly connected to. Adding something directly to the ceiling will have minimal effect unless it is a very, very thick sound dampening material. suspending the ceiling below on sound isolation clips, insulating the joist spaces in the floor and using a sound dampening underlayment under your kitchen floor would be the best way to do it but that requires tearing open everything. You may be better off getting the dog to give up the bone in favor of plush toys.