Kink oscillations of solar coronal loops are frequently observed to be strongly damped. The damping can be explained by mode coupling so long as loops have a finite inhomogeneous layer between the higher density core and lower density background. The damping rate depends on the loop density contrast ratio and inhomogeneous layer width. The theoretical description for mode coupling has recently been extended to include the initial Gaussian damping regime in addition to the exponential asymptotic state. We consider three examples of standing kink oscillations observed by SDO/AIA for which the general damping profile (Gaussian and exponential regimes) can be fitted. Determining the Gaussian and exponential damping times allows use to perform seismological inversions for the loop density contrast ratio and the inhomogeneous layer width normalised to the loop radius. The layer width and loop minor radius are found separately by comparing the observed loop intensity profile with forward modelling based on our seismological results. This seismological method allows the internal and external Alfven speeds to be calculated, and be used for further calculations such as improved estimates of the magnetic field strength or the phase mixing rate.