Split ladyfingers in half lengthwise. Brush cut sides of ladyfinger halves with Kahlúa; arrange, cut sides up, in bottom and up sides of a 10-inch springform pan. Cover with plastic wrap.
Prepare whipped topping mix according to package directions using 1 cup cold milk and vanilla extract.
Combine sugar and cocoa. Beat sugar mixture and cream cheese at medium speed of a mixer until well-blended (mixture will not be completely smooth). Add 1/2 cup milk to cream cheese mixture; beat well. Gently fold in whipped topping. Pour mixture into prepared pan. Cover and freeze 4 hours or until firm.