With such an infestation, you really need to blitz them. Being a pyromaniac I prefer one of those weed wand flame thrower things. Very efficient but there is a slight risk of setting the place on fire
. We then drench the house in Creosote, into every nook and cranny. You do need more than one house so you can let the creosoted one dry out. We had no luck with poultry shield or any of those products. We do though sprinkle DE around the place, especially nest boxes. We also douse the hens in it every now and then.
For the predators, it wouldn't be a rapid response as they have to work up their numbers, and nor would they ever clear the whole place. I think, but don't know, that DE would bump them off too.