Which ones for me then? 12 foot wall, corners available if needed, firing down 15 feet. Would require a complete change of amplification paradigm though.
K's are a nice size.
I had ANE's in a room approximately your size - they worked well, although I preferred not to have them jammed into the room corners.
I don't think however that K's, or indeed ANE's, will give you the resolution and bass at low listening levels that you are looking for. I prefer LV OBX or IBX (either R2 or preferably RW) for that and have had a pair of OBX-R2's in my current room for about 18 months, though they were not my main speaker. But I think there is no getting away from, at least in my case, the empirical observation that if you really want to enjoy visceral bass at low levels then you need a speaker with a big driver.
My current room is 13 1/2 by 15 1/2, extending into a dining room (typical Edwardian terrace with the wall knocked through to the rear reception).
I listen to a 3-way speaker (the discontinued ART Impressions but with Kondo drive units and x'overs and a 12" Volt bass driver as used in the original), with the speakers placed along the long wall either side of the fireplace. I do intend to try them at some point firing down the room.
It is mid-field listening, the presentation is very well integrated and I do most of my listening at low levels late at night. The bass clarity and foundation remains at any level. One big help was placing the speakers on Townshend Stella stands so zero issues with bass and clarity gains across the whole spectrum.
I believe that full-range speakers can be accommodated, with a bit of care, in moderate sized rooms.
My current speakers will be moving to another home, so I am in a similar position to you in that I will be looking to achieve the same objectives you gave. There is another LV speaker under development which would of course work well with my current amplifiers.
But rather than considering an amplification change you might consider PMC, which I have been impressed by, and also the ME Geithain range. I am hoping to listen to the 901K at home at some point, having very much enjoyed listening to it at a dealer's. It is an active studio monitor that doesn't conform to monitor stereotype (dry, clinical, hard etc.). It is a 3-way with the drivers configured to present a point source around a 15" bass driver. They really 'breathe', have tremendous clarity, extension and fluidity, and simply sound very natural. I did ask to listen to them at low volume and it was very satisfying. They have a cardioid arrangement for the bass so that rear energy does not build up, and should therefore be less sensitive to the room. The recommended listening distance is 6-12 ft. They do also have other smaller speakers in the range, which I have not heard, but apparently share the same sonic characteristics.