Would I be able to rent a nice house on the West Side for $1,800-2,500 a month?
Absolutely. I pay just 1000E a month for a large flat (2 bed 2 bath, garage, large terrace, pool, tennis courts, huge living room and sun room) in an expensive neighbourhood of Madrid proper. Going just outside the city will be cheaper so you will certainly be able to find something nice for your budget.
Just be patient. Flat-hunting is an exhausting and frustrating process, but there are good places. Just be ready to grab it when you find what you want. Don't say that you'll think about it since it will be gone when you go back.
I've lived in Madrid for almost 2 years now (originally from NYC) and I love it.