They are over-complicating it.
Build the fence. Secure the border. That shuts off the new influx.
Enforce the law. Let the chips fall where they may. There is no timetable, you just shut off the influx and enforce the law and time will take care of the rest.
Stiff fines and jail the employers. Stiff fines and jail for the landlords. Things will change, very, very quickly. “Self deportation” is an excellent concept with an unfortunately unconvincing name.