Here is the item again:
A student fascinated with all things ancient
Egyptian. (A)
Kemette squinted at the computer
screen,
(B) hoping
to find the information she needed
(C) to finish
her research paper on Old Kingdom art.
You correctly realized that the period after Egyptian created a fragment. A student fascinated with all things ancient Egyptian is an appositive, which cannot stand alone as a complete sentence.
To fix this fragment, you chose to do this:
A student fascinated with all things ancient Egyptian,
Kemette squinted at the computer screen,
hoping to find the information she needed to finish her research paper on
Old Kingdom art.
You have correctly employed Punctuation Rule 9:
Appositive + , + Main Clause.
Play on!