Very often in this show the murder plots are ludicrously convoluted with such a high risk factor that the killer would surely have been better off just hiring a hitman. This one was quite brilliant in its simplicity - so brilliant, in fact, that there was literally no evidence to tie the perpetrator to the deed. Once again, the show breezes by this obstacle with the DI simply narrating what happened with no objections.
I will say that this is one case where I could actually imagine the murderer breaking down and confessing out of guilt and grief. A little bit of work to modify the usual formula (e.g., the killer comes into the station and admits they did it, DI Mooney says he knows and walks the person through how and why they did it) would have been useful here. In any case, the specific circumstances of this murder are why I'm more willing than usual to accept the tired trope of the murderer's pseudo-confession; it's just that an actual confession - the kind that would be admissible in court - could easily have been worked into the story in a believable way.