Best Integrated Amp (for me)

Rotel A series, A-11, A-12 and A-14 are all nice afforable Integrated Amps. If you need digital inputs, the 12 and the 14 have 2ea of opt and coax. I believe the 11 has all rca inputs.

Plenty of good recommendations.  If you have a MM cartridge, consider the Heed Elixir.  Over budget, but you have good speakers and I would advise you not to cheap out too much or you will hear it.  My advice is to buy new so you have dealer support and warranty and spend a bit more to get something you will like a few years down the road. 

A little patience will be required if one isn't listed but look for a used, Rega Elex R.  That speaker likes a little power are needs about 70w of good quality current to sound great.  The Brio is a little shy of what you need in terms of power but will match brilliantly with the Kefs.  

All good choices above.
Personally, I'd pick up a vintage Marantz or Luxman from the '70s.
I own several that I use as described in your opening post, I've not had any issues.

how much volume/power do you need?

Your speakers are low efficiency, only 85 db, which means more power is needed for the same volume level than more efficient speaker designs. I presume you are using the KEF’s in a situation where a lot of volume is not needed. If so, amp power will not be an issue. I am doing well in my small office, low 87 db sensitivity and only 10 wpc tube amp.

For a main system, larger space, more volume needed, many of us recommend trying for higher efficiency speakers, near or above 90 db. Then lower powered amps, especially lower powered TUBE amps can be used if desired.

Less power = less heat; less size and weight; more placement options; and presumably less money.
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Do you now, or might you sometime add a sub?

IF you select an amp or integrated amp, try to get one with pre-out/main in, or a tape loop, or a processor loop (any line level out/in loop). That will let you send the line level signal out to a self-powered sub with it’s own crossover and amp for the low bass. The sub will send the line level without the low bass to your amp’s main in. Now your amp does not need to make low bass, and the less efficient speakers also don’t need to try and make low bass.

I planned on doing that, but made an improper assumption my little Luxman had that, it doesn’t. I’m ok in this small office, but not in a larger space with inefficient speakers. Vintage Luxmans usually have pre-out/main-in, this new SQ-N150 does not.