Before you order a fan, lets better understand why a dead system would even need a fan.
A dead system is often a bad power supply. It’s best to get the system working before you worry about the fan.
Mac mini Mid 2011 Power Supply Replacement
This series also had issues with its logic board. There are two versions ones with just the Intel intergraded graphics engine and a second version with a dedicated GPU chip.
The dedicated GPU models tend to burnout the GPU chip from heavy CPU and GPU processing loads as the cooling was just not up to the task of cooling the system.
So lets first ID your system to see which one you have as you maybe end up needing a new logic board as well!
Plug in the systems S/N here EveryMac - Lookup
crwdns2934105:0crwdne2934105:0
crwdns2934113:0crwdne2934113:0
crwdns2915270:0crwdne2915270:0
crwdns2889612:0crwdne2889612:0
1