Take another look at the original sentence:

Sighing heavily, Melody tried to summon the energy to get to her next class her heavy book bag and the long walk, however, seemed impossible challenges.

You chose to fix it the correct way:

Sighing heavily, Melody tried to summon the energy to get to her next class; her heavy book bag and the long walk, however, seemed impossible challenges.

Melody tried to summon the energy to get to her next class is a main clause. Her heavy book bag and the long walk, however, seemed impossible challenges is a second main clause. A semicolon is an excellent way to join two main clauses that are closely related in meaning.

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