The minimum size range you are looking at is usually 2 bedrooms over here, but (at least with a private landlord) you can sometimes luck out and find a place where one bedroom wall has been knocked out to give a larger living room. It has certainly happened with me. And don't be afraid to check out an apartment in a building which looks distinctly grotty on the otuside - often people will have done up the interiors beautifully, specially in older buildings in well-established areas.
As far as distance from Kowloon MTR station is concerned, 2 - 3 miles gets you out as far as Nam Cheong, Hung Hom, Kowloon Tong MTRs, or across to the western end of HK island. It's very much accepted here that high class private developments will sit cheek by jowl with public housing; you don't get the separation of "rich" and "poor" districts you do in a lot of places. Also, despite the high population density, there really isn't any district where threats to personal safety are a concern.
Still, each little corner of town has its own distinct vibe, so do get your walking shoes on and just use whatever time you can get to mosey around different areas at street level in order to find a place that feels "right" for you (my own personal preference in the radius you specified would be Sheung Wan/ Sai Ying Pun on HK island).
For the initial house-hunting time, prepare yourself to put up with a small shoebox of a serviced apartment for 15K.