The correct way to revise this sentence would be as follows: While she slept, the alarm kept ringing in Jamie's ear. When "while" is used as a conjuction it has two meanings: it describes something that is taking place at the same time as something else or it indicates a constrast.
When you are talking about two things happening at the same time, you should not use a comma if the while-clause does not occupy the initial position in the sentence.
* The alarm kept ringing in Jamie's ear, while she slept
Yet you should use the comma in middle position in the sentence separating the two clauses when the conjuction means the same as although or whereas.
#I prefer The Beatles while my brother prefers The Rolling Stones
I prefer The Beatles, while my brother prefers The Rolling Stones
Regardless of what I've said, you should use a comma after a dependent clause that starts a sentence. Dependent clauses are grammatical units that contain a subject and a verb but can not stand on their own.
* While she slept
That is why the correct way to revise the sentence would be to add the comma after the while-clause.
