The Collaborative International Dictionary of English v.0.48:

Mightiness \Might"i*ness\, n.
   1. The quality of being mighty; possession of might; power;
      greatness; high dignity.
      [1913 Webster]

            How soon this mightiness meets misery. --Shak.
      [1913 Webster]

   2. Highness; excellency; -- with a possessive pronoun, a
      title of dignity; as, their high mightinesses.
      [1913 Webster]

WordNet (r) 3.0 (2006):

mightiness
    n 1: physical strength [syn: might, mightiness, power]