If my studio was broken into and thousands of dollars in valuables stolen, I know who did it, how must I prove it was them?

I live in a downtown area of a big city where there is known to be a lot of crime, including theft and burglaries. Unfortunately, it's very hard to decipher who is and who is not your friend or even a person who you can trust for something pretty trivial such as taking care of your dog for a few hours. So, that's what starts off my story: I was going away overnight for work and my boyfriend needed to take me to my hotel and would stay with me until my flight the next morning. It was about $200 to bring the dog to the hotel so we asked a friend, someone we had recently seen, someone whom we know takes a liking to our dog, to watch him in our studio for about 8 hours. I hadn't known it until we arrived at the hotel but my boyfriend had a bad feeling about all of it. When we returned, our dog was roaming the streets as he knows how to let himself out of the building and several of our valuables were gone, tallying up to over a thousand bucks. We had a note left for us saying that the guy we had watch our dog for us was in jail and to call the number below for information. When I called, it was a man who he said the perpetrator called that evening to come over to my studio. He said when he arrived, he saw the suspect with things of mine that I described to him, walking out of the building. He said he seemed very nervous and told him he was going to "turn himself in" to jail for a murder the cops thought he committed. So, the note turned out to be a blessing but that wasn't the only thing that was left. There was also a cigarette butt in the sink. We don't smoke and don't allow smoking in the apartment so there's really only one answer. It’s the perpetrators. I saved it in a sealed bag but will the cops even bother sending it to a lab to be tested? What are the grounds for conviction in a residential theft or burglary (I know I had let him in)?