It is unlikely you'll find a place to rent in the said amount given the season is picking up in the hills. If not a homestay/lodge, perhaps you could check for hostels.
Short answer: yes — you can find private single rooms / homestays near Kalimpong and Darjeeling for around ₹5k–₹8k/month, but availability and exact pricing vary a lot by season and listing; late Nov–Dec can push prices up, so expect to negotiate or look for long-stay discounts.