I have never heard the K01, but from what I read, it's a significant step up. 2x the # of DAC chips/channel, 4 for the K-03 and 8 for the K-01. Whether it's worth 10K more new, I'm not sure but considering it's about $4K more used, I would say go for it if it's in your budget.
Also, I'm 95% sure that the K-01 and K-03 have an identical feature set (filters, upsample, inputs, volume, etc) since the manual is the same for both units and I did not see any feature call-outs for K-01 only.
As for sound, I have owned the D-07, the D-05 (and P-05) and now the K-03. While the K-03 is still not 100% broken in, the biggest benefits I hear with the K-03 is improved clarity of highs (symbols, lip smacking, snare drum whack, etc), an increase in air around instruments creating more of a 3d effect, tighter and more articulate bass and my favorite - much improved PRAT. The first thing I noticed, even w/o break-in was how the music flowed effortlessly. For a moment, I literally thought some of my well known tracks sounded slower when I realized that was not the case, it's just that much better at PRAT. If the K-01 does PRAT better than the K-03, for me that alone is worth it. It sounds more, dare I say it - "analog". Of course, an external clock may do the same but probably costs more than the upgrade to a K-01. So, long story short, yes, the image density, placement, and size (can be wall-to wall w/the right recording) is outstanding IMO. K-01 would be on my short list if I had the extra $.