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I took a six-month leave of absence. ( is ‘six-month’ a compound adjective as in a blue-eyed lady doctor?) - that’s fine.
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I took a six months’ leave of absence./ I took a two days’/ two-day leave of absence - so are those.
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She took a day’s leave of absence. - that’s fine too.
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She took a day leave of absence. (this one doesn’t seem correct). - your instincts are correct. This is not right. You would need to add ‘one’: She took a one day leave of absence.
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He took a year’s leave of absence. - That’s okay.
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He took a year leave of absence. - You would need to add ‘one’: He took a one year leave of absence.
What about this one: I was granted a six-day/ six’ day sick leave by my doctor. - Yes, without the apostrophe in the second.
All those are acceptable in informal English. Strict grammarians and examiners may insist on hyphens in all the x-day/year/week constructions.