Staying in downtown Osaka is extremely expensive, while staying in the suburbs is quite reasonable. KIX isn’t very near the city anyhow. When I was there, hotels near KIX were 1/4 the price of a downtown equivalent and it’s not expensive to catch a train downtown. (Indeed, less expensive than actually catching one from the airport itself.)

My favorite place to stay in Osaka was Hotel Kinki—a clean, simple business style hotel with lots of street activity at night. It’s in the north part of the city near the Hep5. Very easy to get to/from by train from the Airport. The room had a mild cigarette odor, but then so did the other 2 less appealing more expensive places I stayed. If I go back to Osaka I’ll stay there. Advance reservations recommended.

i personally don’t think that staying in the heart of osaka would be worth it. i’d rather stay in the outskirts and use my money elsewhere. when my boyfriend visited me in japan he stayed at the weekly-inn at makino station on the keihan line. it was super cheap, REALLY clean, and fun. the keihan line runs from kyoto to osaka. kobe, on the other hand is a completely different city. i only went there once. i believe it’s about a 45 minute train ride from osaka, as is kyoto. KIX is right by the ocean so maybe staying near there would be more worth your while. it is not right in the city (it’s an airport, of course). i don’t know if this helped!

i stayed here and absolutely loved it. it’s not that pricey either!
http://www.kaneyosi.jp/english.htm