Best Integrated that you've heard under $5k

Simaudio has some good integrated that shocked me when I learned their price.

Rega has some nice integrated as well.

And NAD was my first integrated and that brought me years of enjoyment.

I really like Simaudio gear also.  They are pretty expensive and I would have a hard time paying their MSRP, but I was able to pick up a 600i for well under your budget.  I've compared it to integrateds from Heed, ARC, Vincent, and Peachtree and it's better than any of them.  I also have a Simaudio headphone amp and it's the best I've heard after owning a number of different headphone amps. 

There's always something better, and "better" is always a matter of taste (unless you're one of those ASR guys), but if you can find a pre-owned Simaudio integrated in your price range I think most people would find it very enjoyable.

Easy... Kinki Studio EX-M1. You would have to spend over five grand to significantly better it. It's only 2400 shipped. Sounds like it should cost 7 Grand

+1 for Raven Blackhawk.  

+1 for Audio Hungary Qualiton x200. If your speakers are on the efficient side, the Qualiton a20i is another solid contender. Excellent class A tube sound.