What kind of budget do you have? I'd be inclined to upgrade everything except the case and DVD burner. After 5 years, it's time.
Considering how cheap a case and DVD burner is, you may just want to take the whole computer and find another use for it. It should still have plenty of life left in it for other purposes.
One option may simply be to upgrade the video card. I have grandkids gaming on machines with similar specs - in fact, one of them did have a 1650 video card. We upgraded that a couple years ago with good results. The only thing you need to do is be mindful of power requirements - that Thermaltake isn't that great of a PSU.
This is the strongest card I think I'd run:
Newegg.com - SAPPHIRE Radeon HD 7750 1GB 128-bit GDDR5 PCI Express 3.0 x16 HDCP Ready Video Card ( 11202-00-20G)
This is also a pretty decent card - in fact, that's what one of the grandkid's systems has:
Newegg.com - SAPPHIRE Radeon HD 6670 1GB 128-bit DDR3 PCI Express 2.1 x16 HDCP Ready Video Card (11192-22-20G)