If you're talking strictly about 1K, then get the Steelseries Luxury Edition Flux Headset.
It's an on-ear design, with nice build and sound quality. Also looks great as well.
There's also a headset called - Gamdias Hebe E1 - which also goes for sale quite frequently on Amazon for 1K. But personally, I'd pick Steelseries.
Apart from that, if you can wait though, the HyperX Cloud Stinger goes on sale on Amazon for 2.4K sometimes. So, if you can save a bit more, wait up and snap that deal - the Stinger would be a huge upgrade over these two!
Agree that a headset is the way to go and you don't need to invest much. A decent set of earbuds or even something like this will be great.
these + headphone cover because some reviews say headband can be uncomfortable after like six hours.
http://www.amazon.com/SteelSeries-Gaming-Headset-Mobile-Devices/dp/B008ZT7X62
check them out, got them for about 2 years (all black version) and i'm very happy with them. Very comfortable and they sound very nice for price IMO. Microphone is on cable which is not the best solution but they are for 17$ right now.
Linus reviewed them while ago:
https://www.youtube.com/watch?v=kx1bRuy3G8M
Edit:
white ones for 13.99
http://www.amazon.com/SteelSeries-Gaming-Headset-Mobile-Devices/dp/B008ZT82IU/ref=pd_sim_147_1?ie=UTF8&dpID=31W0Iq6i-bL&dpSrc=sims&preST=_AC_UL160_SR160%2C160_&refRID=03GJDM05R59SSWDN7RVA
In that price range I would set the Steelseries Flux https://www.amazon.com/SteelSeries-Gaming-Headset-Mobile-Devices/dp/B008ZT7X62/ref=pd_bxgy_147_img_3?ie=UTF8&psc=1&refRID=F8MFA67FMRYPZYDMZT34
EDIT: The white ones are 12.66