Mix flour, sugar, yeast, and salt in a bowl. Combine milk, eggs, and vanilla; add to dry ingredients and mix.
Knead in softened butter until dough is smooth and slightly sticky, about 8-10 mins. Place in a greased bowl, cover, and let rise until doubled, 1-2 hrs.
Roll out the dough on a floured surface into a ¼-inch-thick rectangle. Spread chocolate, then marmalade, over the surface.
Roll tightly into a log, seal the seam, and slice lengthwise. Braid the halves, cut sides up, and tuck the ends.
Place the braid in a greased loaf pan. Let rise until puffy, 30-45 mins.
Bake at 350°F for 30-35 mins.
Simmer sugar and water to make syrup. Brush over the hot babka for shine and moisture. Cool 10 mins in pan, then transfer to a rack.