Right now the coming prepatch is converting Triumph badges to 11.58 Justice Points. This is reflected by the group finder now rewarding 23 Justice Points as a queue bonus (2 Trump badges = 23.16 Justice Points).
This means on average, the typical 3 boss dungeon will reward 57 Justice Points (3 bosses + queue bonus). Heirlooms are presently priced at 2175 Justice Points each for armour. That’s 38.15 dungeon runs for 1 Heirloom.
Presently in WotLK Heirloom armour costs 40 badges, you get 5 per 3 boss dungeon you queue. Meaning 8 3-boss dungeons for a piece.