How are these parameters derived from readings of sensors in an aircraft: angle of attack, slip angle, and airspeed? From what I've learned, the reading comes from 


an angle of attack sensor and a slip angle sensor. Let's call the readings $\hat\alpha$ and $\hat\beta$ respectively. These are of either the weathervane type or the differential pressure type.
airspeed sensor, i.e. a pitot tube. This probe measures the total and static pressure $\hat p_\text{total}$ and $\hat p_\text{static}$, and the airspeed $\hat u$ is obtained from them.


Here's the catch: in real flight, the measured angles $\hat\alpha$ and $\hat\beta$ may depend on not only the true values $\alpha$ and $\beta$ but also other factors which include true airspeed $u$, and the measured airspeed $\hat u$ may depend on $\alpha$ and $\beta$ too. How is this problem solved? How are these sensors calibrated? Are they tested in a wind tunnel where every possible combination of $(\alpha,\beta,u)$ is given to produce $(\hat\alpha,\hat\beta,\hat u)$ and form a lookup table?
They use test flights.

They attach more instruments to the plane and move them outside the influence of the planes own turbulence.

From the known good values of these test instruments they can calibrate the actual instruments.