There's the Polish Cultural Centre (with a bar and a church) at the top of Chapeltown Road, next to the hospital. There are also a few shops with Poland-specific products a bit further down the road towards town
Plenty of good estate agents for rentals in Leeds who all use Rightmove, plus Gumtree Leeds has plenty of adverts. Worth mentioning that houses and flats in the best moderately priced areas (Chapel Allerton, Headingley and Meanwood) are let quickly so you may need to be persistent. Burley and Hyde Park are cheap and mostly rentals but living there requires a tolerance for students.
If you're looking for just a room or bedsit and/or don't mind South or East Leeds (most people who can afford to live in the north of the city) try the Yorkshire a Evening Post small ads. The cheapest accommodation is usually in there.