Several readers asked, apropos this question, why we say “father-in-law” and such at all. The Oxford English Dictionary says that this flows from Canon law, which defined which marriages were forbidden. “[A] brother-in-law or sister-in-law [was], as regards intermarriage, treated ‘in law’ as a brother or sister.”
I don’t think this explains, though, why we usually don’t say “uncle-in-law” or “niece-in-law”; according to this Canon Law source, the prohibition on marrying one’s wife’s relatives extends to aunts, uncles, nieces, and nephews. (If I recall correctly, Leviticus does not forbid marriage to one’s niece, even a blood relation, but Canon Law departs from Leviticus in this respect.)
Maybe the Canon Law rule was different once upon a time, but I have no reason to think so. Moreover, “-in-law” sounds like the sort of suffix that English speakers would extend to other terms even beyond what was authorized by the original etymology. If we can talk about workaholics and telethons, we can talk about nieces-in-law. But for some reason, we don’t.