I understand your dilemma, but putting is before the benefit is not a good idea. In an indirect question, there is no subject-verb inversion:
Normal: you were here (subject, finite verb).
Direct question: she asked: "where were you?" (finite verb, subject).
Indirect question: she asked where you were (subject, finite verb).
In speech, you can't hear punctuation, and so only the tone at the end of the sentence (it goes up only in direct questions) and the word order differ.
There are also indirect questions embedded within direct questions:
- Did you ask her | where she was?
Here the tone goes up because the main clause is a direct question (hence the question mark); and so word order is the only difference between the two. As a result, some people are a bit sloppy with word order in indirect questions in writing and use inversion; but this is not recommended by style books, nor by most educated speakers. But you will see it occasionally, especially in longer sentences.
In your example I agree that putting is at the end makes the sentence unwieldy. The solution would be to move is more to the front, but not before the subject:
I don't know what the benefit is of closing schools when there's a flu outbreak.
I don't know what the benefit of closing schools is when there's a flu outbreak.