Suggested fix
I did the same thing and came up with the following fix.
Basically I could not reconnect the plug back to the mother board. I tried liquid solder (silver based glue) but I think that will only work if it is partially broken or you need to fix circuit boards so that did not work.
I connected the MacMini fan via USB instead. I used an old apple charger cable and cut the usb connector end of with a few inches to spare. then ran a few test on which of the fan wires ran on the two positive and negative usb wires using a usb charger for power. Pretty soon you figure the correct combo of two of the fan wires connecting to two of the USB wires. The USB provide the 5 volts needed to run the MacMini Fan.
Simply put, I have wired the fan directly to a USB port os it runs when the Mac starts up and turns of when you shut it down. Its up to you how you feed the wire through to the outside of the case so you can plug the cable in but I found feeding the wire under the heat sink and drilling a hole the width of the cable. Obviously you need to do this first (once you have sorted the wiring out) then feed the cable through. Solder the two usb wires to two of the fan wires and your done. Just clean it up and tape unused wires down (I cut them off). and your done. I have ran that config for 6 months now with no problem.
Note, The other fan cables not used are simply for controlling but the OS. Just pick the two that operate the fan at 5v on a nice permanent speed.
Cons:
Yes, it is not perfect. Obviously the fan does not run according to the temp controller but Mac OSX doesn't seem to notice or care. I installed SMC fan control to show the temp but it sits at 30-40'c in normal operation and max 85 when doing heavy duty video editing. SMC shows the fan speed as 0 and the temp as it is.
Is the fan running at full and noisy? No I think it runs at about 1/2 to 3/4 speed (permanently). I have another MacMini (Server) and when it heats up it runs loader/faster than my MacMini.
Only thing is I tend to switch it off a a little more when not in use because I am trying to conserve the life of the fan.
Cost to fix: $0 if you have an old USB cable. I looked at the logic board option and tried the fan option first. If you don't like the fan option simply unplug it from the USB and throw it (or re-solder the old wires back to the fan and try another option - if you know if one?)
PRO's: You keep the original fan and the patented Apple fan suction through the front vent (underneath) work perfectly. It looks and operate identical to my other MacMini except for the small cable neatly coming out the back and plugging into one of the USB ports and also the fan runs permanently.
I have never had any over heating issues either.
Sorry, Not sure how to upload pics? Can you only add pics though weblinks?
Also, sorry to disagree with Guf but the MacMini patented fan system only works with the black base on. It draws air through the Hard drive edge of the black base, through the hard drives then across the logic board and CPU then through the fan and most importantly across the heat sink and out the back. You MUST use this system to cool your Mac Mini otherwise the heat sink is not operating properly (or at all) and the CPU is not cooled. You can't simply blow air on the logic board. Google patent apple machine fan or similar. See also image if it uploads. On that note, if you really want to cool down your MacMini use arctic silver heat sink past and sit the mac mini on its side in a Newer technology stand. Drawing in of air works perfectly and both sides are cool. My temp on both my mac minis is around 35-40'c (do the maths) with the MacMini server (with a proper fan) rarely heating up for any length of time.
Silver.