Most of Kira's body was found dead so I assume all that Josuke inherited was half his eyes, half his tongue, his balls, and possibly a couple other insignificant body parts. Josuke also inherited Kira's mind/soul which would be found in his actions and stand, though I'm not sure if it was his entire mind or half. Maybe Kira got a little intimate under the dirt.
Also why do some people call josuke Gappy? I like that nickname and find it funny considering the gap in his teeth, but is a community thing or a nickname that Yasuo (or other character) came up with?