Actually, at 4k the bottleneck is not on the CPU as it moves to the GPU at higher resolutions. Now with your current i7, you wont get as good of fps as you would with a 7700k, but at 4k, the difference is much smaller than it is at 1080p.
In my opinion, for a gaming rig you always want to put your money in a GPU. If you drop a 1080 in your system, you will get better fps than you would if you keep your 5800's and upgraded to a 7700k. And upgrading to just a 1080 would be cheaper than a new CPU/mobo/RAM. So for me, I would go with the 1080 and plan on upgrading your CPU/mobo/RAM sometime in 2018.