I think I would look into renting/borrowing an instrument for DS first, and if that's not an option, I think I would just buy him another one. As you said, not being able to practice/play could affect his grade, and certainly trying to play on a poor-quality instrument will affect his enjoyment of it. If you can afford a new/new-to-him instrument, I would just do that, personally. Then if Relative never follows through, or doesn't do so very quickly, you don't have to stress about it anymore.
If/when Relative finally sends the instrument, it might be possible to sell the one you just bought, or if they keep well, keep both for the next time one needs to be repaired. I personally would just want to get this problem off my shoulders and get DS a working instrument, and anything extra that comes in the mail someday would just be gravy.
If she says anything about you getting him a new instrument, just be cheerful and breezy and say, "Oh yeah, the old one was in such bad repair, the thingie fell off and he just wasn't enjoying playing it. Now it's so great that we have two, he uses one at home and one at school, and he'll always have one to keep practicing with if the other has to be repaired." I mean, she can't reasonably be mad/hurt that you bought him another instrument, when she's taken this long to get you hers. This is a thing he needs for his studies, not a totally optional extra.