My procedure would be:
Connect and measure voltages on the pins of the connector during operation (will require an exactoknife to get onto the wires it looks like :-/
You say 5 pins... and the marketing material says it supports PWM speed control and presumably has the speed output. So that means they've got Vcc, Gnd, Speed output, PWM speed control input.
So it's got only one wire for the LEDs - implying it's a single-wire LED controller.
Take fan, disassemble (this will likely be a "destructive disassembly"), trace out the wiring, draw out schematic.
Be on the lookout for a buck converter or voltage regulator, because the LEDs don't run at 12v, so somewhere it's gotta be making a LED-friendly voltage, or putting the LEDs in series and using a controller that can deal with that.
If you can find a part number for an LED controller on a part, you're home free.