If you're going to use a second computer for streaming you can now get something pretty small with a good cpu thanks to AMD, you just need lots of cores so finding something for cheap with a ryzen 5 1400 would be good, it's 4 cores 4 threads but the only downside is you would need to get a gpu as well as it doesnt have integrated graphics... I'm not sure how well amd's APU's would work for it but that may be an option as well! cpu encoding offers far better visual quality, and almost any gpu would work so you could get something like an old used gtx 960, or 950, or a 1030 from NVidia or a RX 550, or older R5 240 or something from AMD for cheap gpus, but at this point this "streaming" pc may be better than your laptop, buying a new computer just to stream is a little overboard , most people just use old pc's after they get a new one unless they have money to blow,