It is certainly important to know where Kassites originally lived, it is generally believed that they lived in the Central Iran where cities of Kashan, Qazvin (Kasvin), Kashmar, ... were named after them, I don't see any evidence about the existence of a Hurro-Urartian culture in this region, but if they lived in the west of Gutian lands, they could be originally a Hurro-Urartian people, in this case they seem to almost the same as Mitanni.
Persian Jan is cognate with Armenian anjn, the Persian word has a prefix but Armenian one has a suffix, they don't relate to the Indo-European word for race/kin.
No, it's not generally believed that the Kassites came from central Iran, at least as far as the scholarship I have read. It's speculated that they either came from northern Mesopotamia or Eastern Turkey, according to Fournet, or the Zagros region on the border of Iraq and Iran (obviously these theories are not mutually exclusive as these regions border the Zagros, as I'm certain you know). As for Hurro-Urartians in central Iran, the Mannaeans seem to have been a mix of Hurrians, Iranians, and possibly Armenians and possibly Kassites. If you're basing all this on -kash names, Mannaean Zikirti corresponded with modern Kesharvarz/Kasharvar.
If proto-Hurro-Urartian was spoken in Kura-Araxes (which, again, could explain some of the apparent linguistic affinities between proto-Hurro-Urartian and proto-NE Caucasian, as well as high percentage of the J2 y-haplogroup in the South Caucasus), a Hurro-Urartian presence south of Lake Urmia seems even more likely as Kura-Araxes spread well into Iran. Perhaps what happened was, as the Kura-Araxes people spread outward toward Anatolia, Mesopotamia, the Levant, and Iran, they began to develop their own regional dialects and customs. These divisions were heightened when invaders (Proto-Anatolians? Gutians?) pushed them further south, east, and west, and cut them off from each other. The ones in northern Syria became Hurrians, the ones in northern Iraq became Urartians, the ones in SE Turkey or NW Iran became Kassites. Just to be clear: I don't think that anybody thinks that the Kassites were Hurrians, but rather that they were
related to Hurrians, just like the Urartians.
But yes, you're right, a good analogy for the Kassites could be the Mitanni: a Hurrian people with an Indo-Iranian or Indic ruling class, or maybe even the Mannaeans: a Hurro-Urartian people with contacts to (and possibly populations of) Indo-Europeans (Iranians, maybe Armenians, maybe Indics).
I've never heard that word
anjn, but
an is a actually a prefix meaning "without." I believe this case also exists in Indo-Iranian languages, but I'm not sure. Regardless, the root of Armenian
anjn would be
jn. Armenians use
jan (soul, dear) but this is commonly classified as an Iranian borrowing.