Some of these amps (iBasso, e.g.) are not $250 or more, which is the MSRP of the K701.
Regardless, as others have mentioned, high ohm phones like the K701 are going to require a lot of money to be properly powered. My suggestion is, instead of skimping on quality and going for a portable amp much less a budget portable amp, go for a desktop amp that pairs well the with the K701 (you'd have to ask K701 owners for this, because I'm not sure which one to suggest here) and can power it properly, but get it used either from Audiogon or the For Sale forums here. There are some great used deals both on Audiogon and here, and, for the most part, users on both sites are reputable and keep their equipment in great condition versus somewhere like eBay.