When an appliance or tool is broken, the Managemylife web site is a good resource to find out what could be happening and what options are available. I'm sorry to hear about the difficulties your having with your Kenmore Refrigerator's handle. I did some research for you at Managemylife.com and found an expert answer to a similar question that I hope will help until your expert can answer your specific question. I attached the link below.
Thank you for your question and I understand your concern.
Without knowing the model number or being there to see what type screw head it has; I will not be able to say exactly what type of tool and size is needed. It will likely require a very small allen wrench if it appears to have 6 points. I added an image below of an allen head fastener and allen wrenches. You can purchase these at any hardware store or builders supply.