I just received my WA6 and I'm going to vouch for the WA6 pairing as well. It may be slightly underpowered but it would probably make the phones less piercing to the ears of some people.
Since I also have the Burson my feeling of the pairings can pretty much be summarised like this (yes been doing a lot of back and forth between the amps):
Burson Audio HA-160: if you like the sound signature of the K701 just the way it is go for this one, however it will only give you more of the same (keeps the phone to a very analytical and articulated sound, plus puts more extension on both ends)
Woo Audio WA6: if you feel that the K701 is too analytical, bright and bass light and would like to "fix" the sound signature go for this combo, the standard drive tubes included will do this (adds bass emphasis, smoother and lush sound), plus you have the options to roll in tubes later
Another great advice. Thanks. I didn't want to make this thread too focused on amplifiers but I think now the title is asking for it.
So keep your thought coming and I will try to make a summary at the end for more convenience.