It's normal to want to put LEDs in everything, I think we all go through that phase, or never get through it. LOL
Usually, the LEDs in everything grows to seven segment displays in everything and then onto LCD displays...
Anyway...
The best thing to do when trying to re-purpose an existing power supply, or wall wart in this case, and are not sure of the polarity, is to use a DMM to find the polarity of the voltage to be sure that you're connecting the polarity correctly.
You could set this up a couple of ways, one way would be, as suggested, cut the end off the wire(s) of the wall wart, and connect directly to your project.
Personally I prefer to leave the wall wart unmodified for a couple of reasons I will get into later. You can pick up the female end that matches your wall wart, to be able to make a nice little circuit, in an enclosure and plug the wall wart into it nice and clean. This serves a couple purposes, if you ever need to replace the wall wart, you should be able to find the same one or similar that will have the same connection. Also if for some reason you have the wiring hanging in such a way that it gets caught on someone or something as the walk past your work area (among other possibilities) it is more likely that the wire will become unplugged from your fan set-up, than pull it and a lot of other stuff off your work area. Apple actually used this idea in a magnetic power plug for this very reason, someone else used it too, previously, but Apple had better marketing, so...
If you integrated an adjustable voltage regulator into the circuit you could control the fan speed, there are other ways as well, but are more advanced topics.