Take another look at the original sentence:
You wanted to fix it this way:
To delete the comma between desk and her would cause an equally bad problem, a fused sentence. A fused sentence occurs when you have two main clauses joined with no punctuation whatsoever. Mrs. Britton was grading papers at her desk is the first main clause. Her students were slaving over their in-class compositions is the second main clause. The spot between desk and her needs a stronger break than blank space.
You might want to consult the rules for fixing comma splices and fused sentences.