I agree that Mr K admires Emma's older-than-her-years ability to manage her father and the household generally -- this is apparent from their first scene with Mr Woodhouse in chapt. 1.

On a slightly more speculative note, it's also possible that Mr K's sincere regard for Mr Woodhouse could have to do with the fact that he's known Mr W all his life -- and therefore remembers what he was like before his wife died, at which time (although still a "valetudinarian") it's probable that he was less fussy, and showed his kindly, generous side even more often. We see this in Isabella, who lives a happy, active family life with Mr John K, even though she gets more querelous when she's with her father.