Instructions
Load your previously saved game in "Pokemon HeartGold."
Play through the game until you reach the Olivine Gym where Jasmine waits.
Walk up to Jasmine and press "A" to initiate battle.
Defeat both of Jasmine's Magnemites. In a particular burst of creativity, Jasmine has two identical Magnemite Pokemon, who are both level 30 pure Steel-type Pokemon. Use any Fire, Fighting or Ground-type Pokemon to quickly weaken both Pokemon one after the other.
Defeat Jasmine's Steelix, her final Pokemon. Despite its name's descriptive certainty, level 35 Steelix is a mixture of Steel and Ground-type. Luckily, the attacks that do double damage against Steel-type Pokemon do not have adverse affects against Ground-type, so you can use the same kind of Pokemon here: Fire, Fighting or Ground-type. When Steelix falls, Jasmine will give you the mineral badge.