Translation:He has two washing machines, one at home and one at work.
They should probably accept it because that's how many people (esp. foreigners) are going to answer, but "at home" is much more idiomatic. "In the house" emphasises the building too much, and too little the fact that you live in it.
At least in the US, it's called a washing machine or just a washer (as in a washer and a dryer). There's no such thing as a laundry machine.
does מכון get used to mean "machine" in modern hebrew, or is it more like "this is the place where I wash my clothes"? e.g. would I use the same phrase מכון כביסה to refer to a stream where I'm washing my clothes?