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Are you strugling in game play with the i-3 ?
The Xeons are rarely on compatability lists, should work anyway.
And the i3/i5 situation is a fallacy, there will be absolute minimum games where an i5 will perform worse, especially when you consider boost.
Incorrect.
The 6600k base clock is 3.5ghz it boost to 3.9ghz making it 200mhz faster than the 6100.
http://ark.intel.com/products/88191/Intel-Core-i5-6600K-Processor-6M-Cache-up-to-3_90-GHz
I would just close down other apps before gaming.
With your budget your not going to get a satisfactory upgrade.