Take a look at the original sentence again:
Bored with the lecture on comma splices and fused sentences, Jayson started
poking Kim with the sharp tip of his pencil finally
she whirled around and slugged him in the mouth.
You chose to fix it the correct way:
Bored with the lecture on comma splices and fused sentences, Jayson started
poking Kim with the sharp tip of his pencil; finally,
she whirled around and slugged him in the mouth.
Jayson started poking Kim with the sharp tip of his pencil is a main clause. She whirled around and slugged him in the mouth is a second main clause. A semicolon is an excellent way to join two main clauses that are closely related in meaning.