OK, here's what you need to do:

(1) Figure out what exact RAM model is in your PC. Your mobo bottlenecks the speed at 2133mhz (it's Double Data Rate, so what the program tells you is half of what it is), so it may have upside you can't see. If it sucks, buy a faster set.

(2) Buy a Z170 or Z270 mobo. That linked case seems to be big enough for a full-sized one so you shouldn't be restricted to mATX. Don't need to spend too much, just make sure it has decent power delivery and reliability.

(3) Link your Windows 10 license to your Microsoft account. Swapping mobos may cause it to require re-activation, which is easy if you have it linked to an online account but can be a pain otherwise.

(4) Carefully swap mobos. You can cheap out and just get thermal paste to reapply the same cooler, but to properly use the 6700K you'll want a better cooler than the shitty stock Intel, even a pretty basic one and swapping mobos is a convenient time to upgrade. Use rubbing alcohol or this magical shit to clean off the gunk from the original cooler before applying the new one.

(5) Enable XMP in the BIOS. This will allow the RAM (either new or old, depending on what you found in (1)) to run at its full rated speed and not just the arbitrary low limit of 2133.

(6) (Optional) Read and follow a guide to CPU OCing. Stress test with OCCT/AIDA64/Prime95. Increase/reduce for stability. Just a conservative OC is fine, don't need to push it all the way.