Windows doesn't care what CPU is installed. It only cares about the motherboard. ANY CPU that can be used on a given motherboard will use the same drivers and Windows does not discern a change of computer when swapping CPUs, only motherboards. The motherboard on the other hand, could have an entirely different chipset, different storage controllers, different audio chipset, different network adapters and is KNOWN to cause problems when changing from one chipset to another even if it supports the same processors. So when changing motherboards, especially if changing to a different model or different chipset, it is usually either recommended OR necessary to reinstall Windows.
And, so long as you associate your Windows license with a Microsoft account, which you can do by following the instructions below, it wouldn't matter even if you changed motherboards, either.
That Akasa CPU cooler is likely LESS capable than the stock cooler that comes on the i5-4670 non-K SKU, which itself shows to be the same 84w as the K SKU, without, obviously, the ability to overclock. It will probably "work" but it will "work" poorly, it will probably NOT keep you from overheating past the recommended thermal specification of 80°C during boost conditions and it is going to be loud and annoying.
I'd highly recommend getting a better cooler and right now the best low cost budget aftermarket model is probably the same as the normally available one, which is the Gammaxx 400. This would also not require you to install a different backplate as it is a push pin design the same as the stock cooler and your current one, but has the benefit of a much larger 4 heat pipe heatsink. It outperforms the Hyper 212 EVO, so it's performance is proven.
Other models worth considering are the Arctic eSports duo and the Thermalright True Spirit direct 140. All of which should be available for less than fifty bucks, with the Gammaxx 400 being around 30 bucks right now which is about five to ten bucks more than it normally goes for but is to be expected with current market conditions.