Credit for the most home built monitor I have ever seen.
This must be a hobby, as the time taken to do this is much more expensive than just buying a monitor.
Fans are typically designed to run on 12V.
You can down volt them for silent operation, but most have a minimum voltage of around 5V.
At 5V, it may or may not start.
I notice your controller board has a power input from a plug pack.
Is this DC 12V?
If so, could you solder to the pins where this input is soldered to the PCB?
Alternatively, could you modify the lead from the plug pack to power the fan?