(Kouki) can be divided into two syllables "Kou" and "Ki", and each sound can have many corresponding kanji. Thus there are literally hundreds of possible combinations that can be read Kouki. Among them, yes, (Kou) is a common kanji meaning "happiness". is a less common variant. The second kanji that corresponds to "Ki" is arbitrary, but the one you chose ( = "shine/glow") seems reasonable and common to me.