Well, it would be a shame to do a new build when your current machine has a lot of life left in it, I mean my 2500k is still going well with its R9 290. I am of course assuming nothing else is hampering your machine like temps or something.
So, you want to stream, something I am not too familiar with but I would guess you could organise a cheap "streaming" PC to take off the work load from your gaming PC. I am sure that would be a cheaper alternative to building a streaming capable gaming rig. If however you wanted to do the gaming and the streaming from a single rig then much has been said about Ryzen CPUs and their ability to pull off streaming and gaming at the same time.