Why does my GAIN remain constant after a few cycles?
$P$ is the state covariance matrix, not the Kalman gain, though the Kalman gain $K$ is usually calculated using it. 

It is normal in simple Kalman filters (ones that do not vary with time) for the state covariance matrix to stabilize after a while so that it is constant.

See, for example, the plots in Figure 3(b) and 3(c) in this PDF which show how both the Kalman gain $K$ and the state covariance $P$ go to a constant value after a relatively short space of time.

Because this happens, the Kalman filter evolution is often ignored in practice and the steady state version is what is implemented.