How was kira caught?

Answer 1

During the fire when Light killed Takada by suicide in a fire with a small piece of the notebook. Mikami unaware of Kira's actions goes to his safety deposit box to grab the death note to kill Takada thus changing his schedule. This moment makes Near and Stephen Gevanni notice the possibility of a fake notebook other than the fake notebook Mikami had all along. So they switch the notebook with the one Mikami put in his safety deposit box so that during the warehouse scene Mikami tried killing the SPK members with the fake notebook which did not work thus proving Light guilty. Also during the episode before the finale Light says "Near I win" during the the 36 to 37 seconds when Mikami wrote the names down in the notebook. And Mikami even called Light "God" since he gave him the notebook. However if it weren't for Mikami's mistake then Light would have won so Kudos to Light for being intelligent