FM2 is a great choice if you/your friend were looking at UP TO $600 begrudgingly. If you want a $600 computer that will perform well for a while, I'd go either FX4300 or i3 (I'd go i3). With an FM2 build (A10-5800k, preferably) you can build a much cheaper system that will easily max out LoL and will play that level of games well for quite a while. Heck, an A10-5800k will be able to play most modern games fairly well....even if you have to turn off your "Ultra" settings. An A10 will be a fantastic all-around chip for general-use computing and light gaming, and will be a good chip at some "current" gaming.
Now, as far as an FM2 build goes: keep your case (I like the Source 210 more, but it's the same price), keep your RAM. AMD APU's like fast RAM....big time. Your PSU and HDD look good. You can even get a cheaper PSU as you don't need 400W.Here's a build.
If there's no intention to overclock, you can go with a much cheaper FM2 mobo. I mean, the A85x ATX motherboard is totally unnecessary if you don't want to overclock. If you don't want to overclock, you can get basically any mobo with an FM2 socket (A65 or A75 chipset) and you can knock about $50 off that build.