In theory someone competent at laptop electronics and good
with a soldering iron could probably fix the mainboard or
tap into the power rail to make the fan run. If the fan
shorted out it might merely need a surface-mounted fuse or
transistor replaced. It is uncommon to find someone with
this level of ability so for practial purposes it would
usually require replacing the whole motherboard.
However since you appear to have a bit of electrical ability
having already probed the power socket with a meter, if you
can follow the circuit backwards, probe till you find the
point where the power rail isn't yet regulated for fan
control, you could solder a jumper wire to that, and another
to ground, perhaps also putting a resistor in series to
limit current some since you probably don't need the fan at
full speed, and power the fan directly that way.
There is probably enough margin in the circuit to provide
power for both fans, but wired in parallel like this both of
them will spin much slower, perhaps not even starting at all
considering how slowly many laptops spin their fans at
anything less than high load/heat levels.
However, some aren't just two stage high/low RPM, rather
they gradually increase RPM based on temperature so it is
possible the fans running at lower RPM would cause higher
temp but that higher temp causes the fans to speed up in
response until there is an equilibrium. That "might" keep
the laptop cool enough, "IF" that is what the circuit is
like.
It's worth a try, but if that won't work I'd do what I
mentioned above, try to trace back the circuit or tap into a
power rail elsewhere. If you can locate the switching
supply for the processor that is another area before which
you probably have access to solder pads on some surface
mounted capacitors at which point a jumper wire could be
soldered, but I don't know if electrical noise from having a
fan connected there would be problematic for proper CPU
operation. Probably not at that point in the circuit, but I
mention it just in case it is.
Generally speaking, anywhere you probe and find enough
voltage per the fan's specs, where there is a larger amount
of copper or a larger capacitor, will be part of the circuit
capable of supplying enough current to power a fan.
There might be other options for powering the fan as well,
if all else fails you could try running a lead to the
battery pack input, putting a transistor on that used as a
switch activated (biased) by consuming only a little current
from the other fan's power lead when the other fan turns on.
You might want a capacitor between the other fan power lead
and ground to smooth what is probably a PWM control circuit
from the other, working fan. Here's a basic example of a
transistor as a switch, though you might need something with
more current capability than a 2N3906 depending on the fan's
current rating. If you cannot find the fan's current rating
I would assume something with a good margin like up to 500mA
though it's probably only half that.
http://www.rason.org/Projects/transwit/transwit.htm
