Did the League of Villains kill Kai Chisaki?
While his current state is "unknown", we can assume that the League of Villians did not kill Kai Chisaki.  


  In Chapter 160 of the VIZ translations, Overhaul himself, asked Tomura if he (Tomura) was going to kill him (Overhaul), to which Tomura replied no, and that he thought up of something 'worse'. 


Regarding that orb you mentioned, Atsuhiro was holding it over Overhaul but he did not compress him, he just compressed part of Overhaul's arm, severing it as seen below. 


   


Later on, we see Tomura saying to Overhaul to 'sit back and have a nice life' after severing his other arm, and robbing him of his quirk, which Chisaki relied heavily on. Overhauls expression showed fear at the thought of spending his life with no Quirk, which as the wiki noted, is ironic, since he spent his life eradicating Quirks from people. 


   


Finally, the first panel of the last page of chapter 160, shows something already burning in the background but on the second panel, we still see Overhaul alive with no hint of him being burned, with the same expression of fear as seen in the previous page. This confirms the fact that Tomura wanted Overhaul to suffer first, in revenge for everything he had done against the League of Villains.