condition={Subscript[K, 1] <
  1 || (Subscript[K, 1] >=
    1 && (Subscript[K, 2] < 1 - E^(-2 v1 + v2) ||
     Subscript[K, 2] > -E^(-2 v1 + v2) + Subscript[K, 1]))}

{Subscript[K, 1],Subscript[K, 2]}={1,-3-Exp[-2 v1 + v2]}

FindInstance[condition, Array[Subscript[K, #] &, 2]]