Best Integrated Amp (for me)

The Parasound NewClassic 200 Integrated is really nice. 
I got one 2 months ago and have now sold off my preamp and power amp....no regrets.
Its very musical, outputs a clean 110 wpc @ 8 ohms, mm/mc phonostage, bass management, a good burr-brown DAC and a back-lit remote that gives you full control over bass, treble, balance, sub-gain, even a sub on/off. I do a lot of low-level listening so the tone controls can come in handy. 
Parasound has been experiencing component shortages the last few months making this integrated amp almost impossible to find. It's supposed to be back in stock by mid-late Summer. In the mean time the one for $900 that Bkeske mentioned is a great deal.

I'm going to go ahead and suggest what I bought 2 years ago and it is still giving me great sound.

The Vincent SV-500 Hybrid Integrated. 

Simple looks in Black or Silver.

Slightly above your suggested budget but worth it in my opinion. 

The Lyngdorf TDAI 1120 is worth adding to your list. It has a phono input, analogue input, plus Internet radio and streaming AND Room Correction! It sounds fantastic for the money and will optimise the speakers for your room - no brainer 👍

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.