If you'd prefer a straight cable, go for the 80 Ohm. That will be an easier load to drive than the 250 Ohm. The K1 should be able to power the 250 Ohm fine though, if you'd prefer a coiled cable. As I say, they are efficient headphones, despite the 250 Ohm rating.
Ohm ratings do confuse a lot of people, as it's not straight forward. Generally 32 Ohm headphones are seen as easy to drive or power because of the low Ohm rating, and that is generally the case as 32 Ohm rating is common for portable headphones where people just wan to use them with an MP3 player, etc. A lot of people see that any headphones that are around 250 or more Ohm need a more powerful amp. That isn't always the case though, especially the DT770 250 Ohm. They are fairly sensitive or efficient headphones, and the higher Ohm rating doesn't mean they need more power. AKG K702 are 62 Ohm, are insensitive and inefficient. Much lower Ohm rating, but require more power. Many people just go by the Ohm rating and assume the DT770 would be harder to power than the K702.
K1 wouldn't do a great job with the AKG's I would imagine, being a more difficult load, but the DT770 250 should be no trouble. The 80 Ohm version would be an easier load again for the K1, but the difference between the 80 and 250 DT770 isn't going to be that much really, when considering how well an amp can power them. The difference between either DT770 and the K702 is going to be far greater.