Although & Despite

I voted despite because, in order to use although, you would have needed "[...] she was feeling sick"; therefore to use although you need a verb AND a subject too (this is only my opinion: I made my choice just because it sounded good to me).

from Oxford Learner's Dictionary

Usage note: although / even though / though
You can use these words to show contrast between two clauses or two sentences. Though is used more in spoken than in written English. You can use although, even though and though at the beginning of a sentence or clause that has a verb. Notice where the commas go: Although/Even though/Though everyone played well, we lost the game. ◇ We lost the game, although/even though/though everyone played well.
You cannot use even on its own at the beginning of a sentence or clause instead of although, even though or though: Even everyone played well, we lost the game.


In the given sentence though, "although" could not be used because it would not be showing contrast between two clauses. "Despite" does the job instead.

There's lots of useful advice here on using despite (preposition) before noun phrases and although (conjunction) before clauses and that's all correct. What is clear is that despite is undoubtedly correct here.

However:

• these two words do not have exactly the same uses
• although can be used correctly here

If I was going yo use "Although" I would probably say:

Although she felt sick she continued working.