Its hard to say. They should be around the rest of the month and the first couple weeks of October. They come in waves. Last weekend was a bust. We caught some Trout but no Kings. The water went from 69 Labor Day weekend to 46.5 this weekend. It almost got too cold. So we are in a little bit of a funk right now, but that could change in a day. To answer your question, there should be something going on for the next 4-5 weeks, but day to day is tough to predict.
Back 15 years ago when the DNR were stocking a ton of Kings, the run would last 8-9 weeks. In fact my father took his personal best around mid November. Things are different now. Stocking numbers are down and so the volume just isn't there. Here is what happens most years. The Kings pile into the harbors... then they wait for a good rainfall and flow in the rivers and they move upstream and leave the harbors. Then it becomes a tributary fishery. This year the rivers are very high already so there is nothing holding them back. It would not surprise me if there are kings way up the Milwaukee river already.
That said, the Milwaukee harbor has a unique situation. Because of a conflict years ago (that's a whole 'nother story!), the Kings are planted in one area of our Harbor and not in the tribs. This really helps keep them in the harbor longer. Its amazing how they can find the exact spot where the truck dumped them in 4 years ago....