From my experience the K701 benefits from two things : a really good musical source, and tubes somewhere in the system. First, it's a very transparent headphone - any coloration or detail before it will really be audible on these. As for tubes, get a hybrid amp, or get a tubed DAC or CDP and match that with a good solid state amp. Basically a good source will sort out and give a healthy channel separation to simulate the left and right channels, plus the tone of an electric guitar especially when you're missing the sound of a really good guitar amp (Marshall, for example); tubes in the amp acan do that too but my personal preference is a tubed source with a solid state amp.
My favorites for the K701, from the few I've listened to, are:
1) Meier Corda Cantate : USB DAC may be basic but it's very smooth and reasonably detailed - with an entry-level dedicated CDP like a NAD C515 and cheap cables (as in the ones that are bundled with DVDPlayers) the built-in DAC sounds better. I swapped custom high-purity copper cables and you can really tell they're much louder than the USB DAC, which most likely wasn't producing a 2v signal for the amp and takes some power away from the single PSU. That's just the DAC - the amp is a clean, detailed, and reasonably powerful amp that scales well with other sources. My favorite so far is a Shanling SCD-T200 and the Cayin CDT-23, but I don't have that much money for either.
This is discontinued but you can look up any feedback on the xMove series; the DAC is similar and the sound signature, clean and (relatively) powerful output of the amp might get results similar to this.
2) Non-oversampling TDA154x-based DAC (ex Lite DAC-AH) and the Little Dot 1+ or MkII : Really smooth, fantastic imaging, 1+ has a bit more grunt, MkII a bit more rounded and laid back but it's no slouch. DAC-AH now has USB versions, as well as similar designs like the TeraDak.
3) Ibasso PB1 : lots of power and current even on single-ended output, used an iPod 5g and Cowon S9 though.