This day in age you can stream with your GPU with Geforce experience/shadowplay,no need for weak but plenty CPU cores,also LoL relays heavily on single thread power and the live streaming will probably reduce your ping a lot,no matter the CPU.
A skylake I3 with good DDR4 ram will be the best option for you while staying close to your budget.
The build you made will also stream well but it will waste a lot of it's power on doing nothing,or doing something the GPU does much better,you will get much higher FPS in Lol with the i3 though.
[video="https://www.youtube.com/watch?v=-wek5XoyRRU"][/video]