Looking for a CPU upgrade for streaming

[CmdData]

Budget (including currency): ~200-250 Euro

Country: Germany

Games, programs or workloads that it will be used for: Streaming Apex Legends, Cities Skylines, older AAA titles from a year ago at 1080p60

Other details (existing parts lists, whether any peripherals are needed, what you're upgrading from, when you're going to buy, what resolution and refresh rate you want to play at, etc): 
I am currently running an already overclocked i5-6600k@4.20Ghz and while I'm streaming specifically Apex Legends, at all low settings my CPU is absolutely pinned and after maybe an hour or two of playing I notice considerable frame drops every few seconds to the point where it's literally unplayable. I have a GTX1060fe and 16gb of ddr4 ram, and stout bequiet dual-tower air cooler which has had no issues keeping my existing cpu cool. 
Bottom line is that I'm looking for an Intel cpu that fits my existing motherboard socket for the above mentioned price that can relieve my suffering i5. Any tips or tricks for the time being would be welcome!

 

[ki8aras]

6 minutes ago, CmdData said:

Budget (including currency): ~200-250 Euro

Country: Germany

Games, programs or workloads that it will be used for: Streaming Apex Legends, Cities Skylines, older AAA titles from a year ago at 1080p60

Other details (existing parts lists, whether any peripherals are needed, what you're upgrading from, when you're going to buy, what resolution and refresh rate you want to play at, etc): 
I am currently running an already overclocked i5-6600k@4.20Ghz and while I'm streaming specifically Apex Legends, at all low settings my CPU is absolutely pinned and after maybe an hour or two of playing I notice considerable frame drops every few seconds to the point where it's literally unplayable. I have a GTX1060fe and 16gb of ddr4 ram, and stout bequiet dual-tower air cooler which has had no issues keeping my existing cpu cool. 
Bottom line is that I'm looking for an Intel cpu that fits my existing motherboard socket for the above mentioned price that can relieve my suffering i5. Any tips or tricks for the time being would be welcome!

 

im gonna be honest with you, by choosing not to upgrade your motherboard youre limiting your performance increase to a best case scenario of around 10-15% by going with the i7 7700k and heavily overclocking it

 

you didnt really ask for it but a workaround worth trying is switching to gpu encoding for your livestream,

 

other than that you can try finding some deals on ebay for something like a ryzen 7 2700 and a budget b450 board (ryzen 7 2700's go for around 150 euros and b450 boards for around 50-75 euros)

 

(im mentioning the above combo because it fits your pricerange)