Maybe he spilled the beans prior to being read his Miranda rights and after being mirandized he clammed up and asked for a lawyer, and if so does that initial statement become a confession?
That is very speculative and I'm quite sure did not happen here. They interviewed him for hours, and he even took them back to the crime scene. These cops are not stupid enough to have not mirandized him before all that.
Even so, the larger point is that having the murder weapon is not required to get a conviction. You need proof beyond a reasonable doubt that the person is guilty. You can have that without proof of every possible detail.