What is the difference between Kalman filter algorithm and stationary Kalman filter algorithm?
I’m not sure what you mean by the stationary Kalman filter, but it seems to be what I would call the steady-state Kalman filter.

If that is the same thing, then you just solve for the Kalman gain at $t=\infty$ and apply the normal Kalman filter equations.

I’m not sure you mean “the Kalman gain goes to infinity” in a well-posed problem.