For Kahlua simple syrup:
Add the granulated sugar and water to a small saucepan over medium heat and whisk until the sugar is completely dissolved. Bring to a boil, then remove from the heat and add the liqueur.
Stir to combine and allow syrup to cool completely.
For Biscuit Roulade:
Preheat oven to 375 degrees F. Butter and flour a 15 by 10 by 1-inch jelly roll pan. Line the bottom with parchment paper and butter and flour the paper.
Combine the flour, baking powder, and cocoa in a small mixing bowl, set aside.
Beat egg yolks in a medium mixing bowl with an electric mixer on high speed until thick and lemon colored, about 5 minutes. Gradually add 1/2 cup of the sugar, beating on high speed until the sugar is dissolved. Thoroughly wash and dry the beaters.
Beat egg whites in a large mixing bowl with an electric mixer on medium speed until soft peaks form, about 5 minutes. Slowly add the remaining 1/2 cup of sugar, beating on medium speed until stiff peaks form.
Carefully fold the egg yolk mixture into the egg whites, using a rubber spatula . Once all of the egg yolk mixture has been incorporated, sprinkle the flour mixture over the egg mixture and fold in gently, just to combine.
Spread the batter evenly into the prepared jelly roll pan . Bake until the cake springs back when lightly touched near the center, about 12 to 15 minutes.
Lay out a clean, thin cotton kitchen towel on a work surface; dust the top with confectioners' sugar and set aside.
Immediately after removing the cake from the oven, loosen the edges of the cake from the pan using a knife.
Turn cake out onto the sugared kitchen towel, parchment paper side up. Carefully pull off the parchment paper.
Fold towel over the longest side of the cake and roll up the cake with the towel inside.
Cool on a rack with the seam side down.
For Coffee Buttercream :
Bring 1/2 cup of water to a boil in a small saucepan over high heat. Remove from heat and add the coffee grinds; allow to steep for 5 minutes. Strain through a coffee filter into a small bowl and set the coffee concentrate aside.
Mix the remaining 1/2 cup of water and sugar in a heavy 2-quart saucepan over medium-high heat. Clip a candy thermometer to the pan and heat to 240 degrees F, soft-ball stage , 18 to 20 minutes. Keep the edge of the saucepan clean by brushing it with a pastry brush moistened with water, as needed.
Once the temperature reaches 240 degrees F, remove the candy thermometer and remove the saucepan from the heat.
Beat the egg whites in a large mixing bowl with an electric mixer on medium speed until soft peaks form, about 5 minutes. Gradually pour the hot mixture, in a thin stream, over the whites while continuing to beat for 2 to 3 minutes. Scrape the sides of the bowl as needed and continue to beat until a meringue forms.
Add the butter all at once, and beat on medium speed until incorporated. Increase speed to high and continue beating until smooth and shiny, about 8 to 10 minutes. Add the vanilla extract and coffee concentrate and mix for 1 minute until combined. Set aside.