Think you need to change mechanics. The reason the light is still on and the car is running badly is because the second lambda isnt reading any emmisions at all. The second lambda is used to determine the efficiency of the cat and to validate the first lambda to some degree.

As for over 5k, it may not open at a constant 5k as it may be trying to save fuel. To test, give the throttle a flick to simulate acceleration and see if it opens. Does sound like the ecu isnt able to determine whether it is under load.

My advice really is to go somewhere with a rolling road and a diagnosis computer to see what is going on. Shouldnt cost much.