See this list of cpu supported by your motherboard.
http://www.asrock.com/mb/AMD/A785GM-LE128M/?cat=CPU
But still i recomended to get at least 4 core cpu. My phenom X4-B55 (unlocked from x2-555) oc 3,6GHz with stock hsf can still ultra on games such as the division on 1280x1024 with 4x2Gb crappy ram and 7950 (R9-280) oc 1050/1400. So rather than you use the money for aftermarket cooler, you better use the money for 4 core amd cpu with stock hsf. Just my suggestion though.
Make sure to get 2x2Gb ram, although i strongly suggest to get 2x4Gb for longer usage in games.
For gpu, rather than using amd's motherboard's onboard gpu, have you considered using lower end of intel series?
But then again I didn't know your total budget. Thus i could only recommended at least get a gforce 750 or 750Ti. These gpu are known for best performance/power usage ratio (electric bill). Yeah, its a US$90 gpu but it will benefit you in the long run. With 4 core cpu and GTX 750Ti, your girlfriend pc will become lower mid gaming.
Not to offend you, but your original build is not even considered a low gaming pc. Its an office use pc.
Good luck with your build.