It is my personal opinion that the K1Ks are less amp-dependent than many posters imply. I have tried both my sets with a variety of amps, both tube and SS, and found that most amps that are capable of delivering 10wpc of clean power into the K1K's relatively high impedance can do a very creditable job.
I have had excellent results with the following T-amps: RWA Signature 30.2, Winsome Labs Mouse, Virtue Audio One, and Little Dot T_150 as well as the modestly-priced but excellent performing nOrh tube amp.
While I am sure many others have hearing that is more acute than mine, I can't believe that I wouldn't be able to differentiate good sound from mediocre sound from the K-1000s!
FWIW, at the last mini-meet I attended, many attendees listened to my K-1000s while driven with my $70 Bravo tube amp and commented on how great this unlikely pairing sounded!