Did some looking and decided your problem is particularly disconcerting. I decided to do a little research to see if I could help. It seems to me that if the situation is as bad as you describe you are going to have to to take a proactive aproach or learn to live with it.
I sincerely hope that this infestation is a seasonal event and will subside soon, however it seems the current information leads to the contrary
I'm thinking you probably already know this, however you wrote that it has "no natural enemies". This is not 100% true. There is a parasitic wasp that is useful in the control of this pest. I don't know how proficient you are at propagating Trissolcus halyomorpha Yang........
A new species of Trissolcus Ashmead (Hymenoptera: Scelionidae), Trissolcus halyomorphae Yang, sp. nov., is described from China. It is a solitary parasitoid in eggs of Halyomorpha halys (St虱) (Hemiptera: Pentatomidae), an economic pest that causes severe damage to pear, apple and other fruit and that also feeds on the leaves and stems of ornamental and forest trees. The new species is the principal natural enemy of the stink bug, with parasitism rates up to 70% and an average annual rate of 50%. Because of its high parasitism rates and other biological features, it has good potential as a biocontrol agent of the pest.
This coupled with the proper application of an effective insecticide may get things under control.
I personnally would be reluctant to use a chemical control unless it was something relativley inert, such as soap. It may take some experimenting to find a way to broadcast a water soap solution so that it does not aspirate but incorporating the use of a low pressure high volume system may be effective.
In the mean time securing the perimeter is the priority. I would emply the whole family for this endeavor and start blocking the entrances. Every last little crack, crevice and hole must be plugged. Perhaps locating, marking and then using an expanding foam migh be one method. I'm sure your neighbors are experiencing the same as you and your family, so perhaps you could work together getting the house sealed and the problem migjht not seem so insurmountable when that is accomplished.
Another helpful link from a University near you.
http://www.virginiafruit.ento.vt.edu/BMSB.htm