Ah, the joy of apartment living. Before I moved here I had the exact same issue with my upstairs neighbors. They would make a racket at all hours, but complain if I flushed my toilet in the middle of the night. Let me start by saying that if your neighbors are such jerks that they won't discuss this with you to come to a mutual understanding, a recording will probably not be much help, and may actually make things worse. That being said, you may be able to get some satisfaction by having the apartment manager hear the recording. Again, this might just escalate things to where you will find your car vandalized, as I did. In my case I was able to get several neighbors to corroborate the all-hours noise, and a few other things the jerks upstairs were doing, and ultimately they were evicted from the place. It was an ugly, drawn out ordeal that I wouldn't wish on anyone. Even then, I ultimately went into unneeded debt to move out of that $#!+hole.
Now as far as recording goes, you should be able to get a usable for your purposes recording with what you have, and microphone placement will be the key. If you have access to a condenser mic it may pick up the low frequencies better than an SM58, but almost anything should do the trick. A little post recording equalization might be needed, but where you place the mic will be the biggest factor. The hardest part will be the time it takes to begin the recording once the noise begins. That's where I had a problem. It took weeks to compile enough audio to show the extent of the noise coming through the ceiling. It was a real PITA.
In the mean time I would talk to your other neighbors to see if they are having similar issues with this family. That was ultimately what did the trick for me.