Well, perhaps they could start a new life away from all the chaos in Mystic Falls. They build a home together in a far - off place and discover new abilities or powers that come with their union. For example, Elena might gain some of Kol's vampire powers in a unique way. They also face challenges like normal married couples, such as learning to live with each other's habits and quirks.
A possible plot for 'Kol and Elena married fanfiction' could be that they decide to open a school for supernatural beings. As a married couple, they use their combined knowledge and experience to teach young vampires, werewolves, and witches. However, they face opposition from some traditional supernatural groups who don't believe in this kind of integration. So they have to fight for their vision while also maintaining their relationship through all the stress and drama.