Stir first 3 ingredients in heavy large saucepan over medium-high heat until sugar dissolves and cream comes to simmer. Remove from heat. Add 20 ounces chocolate; whisk until melted and smooth. Whisk in rum and vanilla. Transfer 1 cup warm chocolate mixture to small bowl; whisk remaining 2 ounces chocolate into mixture in bowl and set aside for topping. Cover; let stand at room temperature. Transfer remaining chocolate mixture to large bowl to use as filling; chill until cold and thick, at least 6 hours and up to 1 day.
Stir water and sugar in small saucepan over low heat just until sugar dissolves. Remove from heat; mix in rum. Cover and let stand up to 1 day.
Preheat oven to 350°F. Butter 9x9x2-inch metal baking pan. Line bottom with waxed paper; butter paper. Combine vanilla and coffee powder in cup; swirl to dissolve coffee. Sift flour, cocoa, and baking soda into small bowl. Using electric mixer, beat egg yolks and 1/2 cup sugar in medium bowl until very thick and light in color, about 3 minutes. Beat in vanilla-coffee mixture. Using clean dry beaters, beat egg whites, cream of tartar, and salt in large bowl until soft peaks form. Gradually add remaining 1/4 cup sugar, beating until stiff but not dry. Fold 1/3 of whites into yolk mixture. Fold in half of flour mixture. Fold in half of remaining whites, then remaining flour mixture, then remaining whites. Transfer batter to prepared pan; gently spread to even thickness.
Bake cake until puffed and tester inserted into center comes out clean, about 18 minutes. Cool in pan on rack (cake may shrink slightly).
Cut around pan to loosen cake. Turn out onto work surface; peel off paper. Using long serrated knife, cut cake horizontally in half. Place 1 half, cut side up, on platter. Drizzle half of syrup (about 3 1/2 tablespoons) over. Using electric mixer, beat cold chocolate filling until thick and lighter in color, less than 1 minute. Spread all of filling evenly over cake layer, aligning with sides of cake (filling will be about 1 inch thick). Drizzle remaining syrup over cut side of second cake layer. Place layer, syrup side down, atop filling; press to adhere. Rewarm topping over low heat just until pourable. Pour topping onto top center of cake. Using small spatula, spread topping to edges, being careful to keep it from spilling over. Chill cake up to 1 day, covering loosely with foil or with cake dome after 2 hours.