I don't know about the cookie recipe, but for the egg whites:
Are you beating in totally clean, dry bowl? There cannot be any oil residue from recent cooking left behind, or water from cleaning. Are there any yolks in it? Make sure it is 100 percent egg white in a 100 percent clean, dry bowl.
Are your eggs ice cold? They whip better room temp. Also, if it doesn't interfere with your recipe, you might try a pinch of cream of tartar.
We're fat chicks, not doctors. Please see your physician before taking advice found on the internet.