Combine chocolates, butter, and Kahlua in a medium bowl set over barely simmering water, stirring frequently until chocolate is just melted. Set aside to cool to lukewarm.

Place coffee in a small bowl and add gelatin. Set aside until gelatin has softened, about 5 minutes. In a medium heatproof bowl, whisk egg yolks until smooth. Place bowl over the pot of simmering water and whisk in coffee-gelatin mixture and confectioners' sugar; continue to whisk constantly until mixture is thickened, light, and fluffy, about 4 minutes. Remove from heat and whisk mixture into melted chocolate. Set aside until cooled to room temperature.

In a separate large bowl, whip egg whites until foamy. Add cream of tartar and, little by little, gradually beat in granulated sugar. Increase speed to high and beat until stiff peaks form. Fold about 1/4 of egg whites into chocolate mixture to lighten it, then gently fold in the remaining beaten whites.

In a large bowl, whip heavy cream to just beyond soft peaks but not quite stiff. Gently but thoroughly fold whipped cream into chocolate mixture. Transfer mousse to refrigerator until moderately set up, at least 2 hours and up to overnight. Use as a filling for cakes or pipe into small dessert dishes and serve on its own.