OK, so let's see. The detective's wife was stalking the guy's sister, so he killed her but then the detective came home and discovered him and they struggled and fell off the balcony together and died. It's all so obvious!
No there's a problem with your time frame..
According to the brother's memory, he discovered his sister all beaten up. He probably then decided to kill the detective's wife (who was a stalker? or maybe a very mean teacher?), which he did.
The detective discovered his wife dead, and he was alone at the time. During his investigation, Tatsumi's name came up, that's when he went out on a walk [to kill Tatsumi] (there's a memory from both of them where they seemed to be walking toward each other in the same walkway..), and that's when they killed each other!
