adding a DAC/Streamer or what

10K is a healthy amount of samolians to heave at this issue.  Still, I recommend being cost effective, in case you want to upgrade.
As indicated earlier I would buy a great DAC and start with a budget streamer, possibly even a PC.  With that budget I would do something much better than the Chord.  So ask yourself what features you want in a DAC.  I use a Bryson DAC3, which besides being a great DAC, as 4 HDMI inputs.  I have many Blu Rays/DVD-A and Blu Rays and the DSD and other high res formats can be sent to the DAC over HDMI (a television can also outputHDMI).  Perhaps this HDMI feature has zero relevance for you, but the Bryson at about 3.5K is still a great DAC.  The Bryston USB inputs are not inferior to their other digital inputs.  Many DACs have an Achilles Heel and that is unacceptable, because many streamers only stream over usb.  I have also owned the Mytek Manhattan, which has no HDMI but retrieves an unbelievable amount of detail.
  Anyway, I would spend 3-5K on a killer DAC, and start slow on the streamer.  A PC optimized with a program such as Audirvana Plus or the Bluesound Node2i.
Another 1-2 K on  a NAS (Synology or perhaps a Melco  N100) if you have a big CD collection to convert to streaming files

What sound qualities am I looking for was an interesting question. I guess my answer is I would like to compliment my component sounds that I already have in my speakers and the amp/preamp. I love to hear the individual components of the music I am listening to and feel that the other equipment I have purchased are able to do so. I like to listen to jazz, classical, and rock. I am not sure that is what you are asking but there you have it. I like the warmer tones and like to be able to hear the bass also.

There’s a plenty of good advice here.  I’ll add my own $.02 as I have a bit of a different view than some.  We all agree on the importance of the DAC.  However, if you have a $10k budget, and given the high quality of your other system components, I’d suggest you can skip budget server/streamer. I had a Node 2 going into an outboard DAC and have upgraded twice.  
The primary difference between a budget unit like the Node 2 and a better unit is in information, and particularly spatial information in the music, not in tonality. Your KEFs are very, very  good overall, and better than most at reproducing spatial information. As you like jazz and classical, it’d be a shame to hamstring them. Since you’re asking for input, I’d budget a similar amount for the server/streamer and for the DAC. Some would say the DAC is far more important than the server/streamer. They may be right, but at least a few of us in these forums have found them to be equally important.
 

@raw33 if you're looking for a DAC that is similar to your existing gear, then I'd recommend researching the following options:

• Prism Sound Callia
• Musical Fidelity M6 Encore (streamer) or M6sR (DAC)
• MSB
• CH Precision C1
• Lumin S1 or X1
• Soekris dac1541

There are more I could list, but those are the ones I thought of quickly off the top of my head.

A thank you!