Kodaikanal hill station would be the best for you.
many foreigners come to this place and be here for around a month or so for very reasonable prices.
do check out my trip and things to do in Kodaikanal
Try Zostel. Their hostel rates are very cheap. They do have their properties in most of the popular places and even in some off beat places. If snow fall and harsh winter wouldn't be a problem for you then you can try for Himalayan Regions where you will get great accommodation at rock bottom prices cause of winter.