I'll preface my remarks by saying I lived next door to crappy neighbors for 11 years until I became convinced that they would outlive and outlast me so I sold out and moved. So I'm sympathetic.
Three things you have to understand.
1 - As a month-to-month tenant (at will) your landlord can terminate your tenancy for any reason as long as it's not in violation of any anti-discrimination laws. All it takes is 1 month written notice per Michigan statute 554.134.
http://www.legislature.mi.gov/(S(0kn...me=mcl-554-134
2 - Your audio recording of your neighbors bickering is a felony "punishable by imprisonment in a state prison for not more than 2 years or by a fine of not more than $2,000.00, or both." See Michigan statute 750.539c.
http://www.legislature.mi.gov/(S(0kn...e=mcl-750-539c
That's why the apartment management wants nothing to do with your recording. It's illegal and would never be admissible in any legal proceedings and you'd probably be arrested if you offered it up as evidence.
You need to destroy it and never do it again and hope nobody reports it to the authorities.
3 - Management knows you are leaving and the crappy neighbors are staying. Your complaint is of little importance to management.
I know you aren't going to like this but my advice is keep your head down, your mouth shut, tough it out, and get out of there as soon as possible. I'm up for a good fight as much as anybody but this is one of those times where discretion is the better part of valor. In a few months you'll be happily ensconced in your own home with the consolation that somebody else is dealing with the crappy neighbors.

