Does Vi Hotheaded equipped with Svellsungur have 6 or 12 might after paying 2 energy and 1 power?

Ruling: 6 Might.

Svellsongur gives Vi two separate instances of her activated ability ([2] [C] : Double my Might this turn.), but each instance requires its own cost payment. Paying [2] [C] once only activates one of those two instances, so Vi's Might doubles once: from 3 to 6 (Rule 432.1).

To reach 12, you would need to pay [2] [C] a second time for the Svellsongur-granted copy — for a total of [4] [C][C]. Each activation is independent and you can activate both in sequence as long as you can pay each cost (FAQ #10207).