Normally besides the smell, you can tell if your system's been smoking if it has red eyes and gets the munchies!
OK... The smell was likely caused by a short. Might be the same short that rebooted the system when you bumped it, maybe not. The bumping caused some component to come into contact with something it shouldn't, most often a 'hot' item being grounded. Could be a bad post holding the motherboard, or a loose wire, or a bit of solder coming loose. The things you saw on the motherboard may not be clues, it's not uncommon to find things like that on any old board, even when it works. I once had a whole line of pots blacken on a motherboard and it kept working for years - no idea what they did other than smell bad.
Sounds like you already re-built the system without the case? If you haven't then do that. Test and see if it'll start. If not, remove all the unneeded connections. Pull all wires from the drives, remove any other cards than video. If no joy, try removing all but one RAM module. Try clearing the CMOS settings. BTW, you can turn on your system with no front panel switches by very carefully shorting the power switch pins on the motherboard with a tiny screwdriver then releasing. Or hook up the switch.
Be sure to point out to your girlfriend that despite the problems you had earlier, she likely broke it! This will make you feel better and might get you extra care. Of course, it might back-fire too.
The best way to test things is if you have some extra parts - like your girlfriend's computer! The most likely part that's dead is the motherboard but it could be the power supply or other things.
If you decide to replace the motherboard, remember that your money may be better spent by upgrading MB, cpu and RAM at the same time. And consider a bigger PSU if needed.
Let us know how it goes with the testing.