Same as by counting of the Omer we do not say extra Kadish
When davening in shul that say aleinu before Ledovid Hashem - say Oleinu first.
Just as during the counting of the Omer we do not say an extra Kadish, so too during Elul when reciting Ledovid Hashem Ori, no special Kadish is added. In shuls where Aleinu is recited before Ledovid Hashem Ori, one should say Aleinu first.
Summary
This shiur explains that during Elul, no additional Kadish is recited after Ledovid Hashem Ori, paralleling the custom for Sefiras HaOmer. The order of prayers should follow local custom regarding Aleinu and Ledovid Hashem Ori.