Anywho, the system won't work without both screens unless you close the circuit so that it thinks both screens are there.
The points aren't all that small and it's really just one wire - you don't need a whole lot of skill to pull this off. Frankly, you might be able to do this with a well-placed piece of tape and one resistor by the looks of it, although that kind of a contraption would probably be dodgy in terms of reliability.
You know what they say - there ain't nothin' that can't be fixed with duct tape. 
Just leave the broken ribbon cable in the wires are for the sound, Just use the headphone jack.
You can always try to make the connection lower down the way, on the nearest available solder point or connector. People connect the resistors to the ribbons simply because it's convenient - you don't have that option, so you'll have to travel down the ribbon and find a different suitable spot.
If you can grab a replacement cheaply, get a replacement. Removing the top screen really works to the detriment of the system in my opinion. Unless you only care about the GBA functionality, I'd keep the top screen.
Oh. Well, in that case you might want to keep it just for spares, but yeah - go nuts with soldering, it's fun!
I do a lot of small repairs around the house and occasionally on my retro systems and gamepads. I'm not terribly good at it (my hands shake a bit) so I'm not very precise, but I make-due.
It's a useful skill and the tools of the trade are really cheap unless you want a superb soldering station (you really don't need one - my soldering iron cost me about $5-10, if I recall correctly ).
