I'm not sure I understand your last about the speed not depending on the medium, or the source.
Me neither anymore;-) (Sorry, BTW, for the long delay, but I hadn't looked at this thread for some time.) Actually, the speed of sound does not depend on the speed of the source, that's where I got it right, i think. But it's constant relative to the speed of the medium, so sound produced in air moving at 100 miles/hour would move faster in the direction the air is moving than in the opposite direction. Thus something like a device making "pings" and measuring the time the sound takes to arrive at some other part of the airplane in principle should be feasible (as far as I can see, but, as you've seen, I can be wrong;-)
The problem I see is that the speed of sound depends on (at least) three things: the temperature of the medium, the pressure and the its composition. Temperature and pressure already get measured (but I don't know if the precision would be sufficient). But you also need to know the amount of moisture in the air (or even if there are water droplets). And that might be harder to measure, especially if flying into and out of clouds at rather high speeds. Where such a device might come handy would be in situations where there are conditions leading to the pitot tubes clogging up with ice, i.e. conditions where there's quite a bit of moisture (perhaps super-cooled water?). And measuring reliably and with sufficient speed the exact composition of the air you're moving through to arrive at a good enough estimate of the local value of the speed of sound might pose a bit of a problem.
One possible solution might be to measure the speed of sound directly and independently by having another "pinging" device perpendicular to the direction of flight (assuming that cross winds aren't too large to mess up everything). I.e., measure, for example, the time it takes a ping from the tip to the root of the wing to arrive at an estimate for the speed of sound at that moment and use this for calculating the speed of air in the direction of flight. Unfortunately, that could make the device twice as expensive (and double the likelyhood of failures;-)