Alex who made his proclamation that Lilitth must die first to live, was simply saying the same line as his dad. He wasn't truly planning to kill Lilitth, quite the opposite he was truly planning to save her. Though he was indeed planning to hurt her until she lost consciousness.
Alex who was full of fighting spirit started to finally truly think of a way to make the loli he was fighting fall unconscious. The things he understood, as he fought the girl were limited, but he was sure that clue lay hidden within that limited information.
The first thing he noticed was obviously her physical capabilities, which go beyond what a human can do. Yet even with her enhanced abilities, she still wasn't able to kill Alex simply because she was fighting like a wild animal.