Both Kawai and Yamaha are good pianos. The Kawai has plastic (believe me, they don't use the word plastic, more like a high tech polymer) wippens which are not impervious to moisture. That's a plus, however, the felt bushings are which means the plastic will not expand to compensate for the wear of the felt. Kawais tend to have a lighter faster action. Many jazz musicians love them for that. Yamahas feel heavier yet the rebound of the hammers are impeccable. I personally like the Yamaha C series better than the RX1 and the Shegaru Kawai better than both. The Shegaru is REALLY pricey and hard to find. A G series Yamaha is good for the average home, but if your into playing Henle edition Chopin etc. etc. you want the C series.
Also, when shopping, you have reconditioned (basically regulated, voiced, get everything to work) rebuilt (new strings, hammers, pins, maybe pinblock, regulated, everything works), and restored (everything replaced, all the action parts, strings, pins, block, case refinished, felt parts), so be careful. The 15K Steinway above is more than likely original old parts just made to work. Restored, you could add 10K to that.
You can get restored Knabes for around 10K. Its a really good piano for the price. No, its not a Steinway, but really holds its own. I'm really surprised nobody mentioned Mason Hamlins, although really close to the Steinway in price. For price, quality and reliability, the Yamaha C is the way to go. Anything under 5'8", you start to lose the sonority of a large piano, but lets face it, most homes are not that big.
One more thing, you can get a C2 Yamaha, 5'8" 10-12 years old for around 12 K, not much more than paying 8-10k for a 5'3" C1. And at that price, mint condition.