what settings are you trying to achieve ?
if its 720p30fps then you wont have to invest as much as opposed to like 1080p60fps
but your internet connection is likely to be a factor aswell with the 1mbit upload you will prob be capped at 540p resolution according to some posts about it on reddit
if you were to buy a completely new system you can try keeping the current pc and use that one as the streaming pc and use the new one for gaming
also whats your budget? if i know what you wanna spend i can give you some builds, your gpu is still fine, its just the cpu thats showing its age when hit with streaming and gaming