When CP first came out I tried DK abilities with and without mighty spec'd particularly unstable flame (Stam morph) and the damage was not improved. It has been some time since CP came out but hopefully that helps. Its only a 3000-6000 gold experiment if you want to confirm for yourself. What will increase fire and poison most certainly is Thaumaturge and Elemental.

There are separate selections that increase poison damage and another that increases elemental damage. Mighty ONLY increases your PHYSICAL damage- as in damage from swords, daggers, 2H, Bow attacks. They do not include the secondary effect, such as poison, fire damage, etc. Poison is covered by Thaumaturge and Flame Damage by Elemental Expert.