Fixability depends entirely on the problem, what equipment you have available to you, and how good you are wielding them :)
The red light coming on is a partial good sign. That LED is basically hardwired between Vcc (the output from the voltage regulator), the switch, and GND. So the battery's okay, the battery contacts are okay, the voltage regulator is probably okay, the switch is okay and that LED is okay.
That does leave plenty of things that may not be okay, of course. While a visual inspection is a great first step - especially in identifying a loose wire, bad solder joint, etc. - at some point you'll have to get out a continuity tester or, better, multimeter.
If you're comfortable with that type of tool, I'd start with the connections from the button daughter board to the main board. If the microcontroller never gets the signal the button is being pressed, not a whole lot would happen regardless of anything else in the chain. It's also one of the more fragile points (others have had the USB port come loose, but I don't think that would affect basic operation - just the charging/programming bit). If something's wrong there, it's probably the solder joint(s) which would be easy to re-solder.
One note - I suspect that if you do start soldering around, your warranty will be void ( I don't know what the status of I-dropped-my-Flex-halp? is with regard to RMA :) )
If those connections are fine, I'd move to the other end of the chain and check the LED. Put 3V (two AA batteries - don't use the 18650, that's 3.7V nominal and the LED doesn't like that very much) across the LED at the wires on the main board (mind your polarity) for just a moment. If it doesn't light up, either the wires or the LED is at fault. Again, might be as simple as re-soldering.
If it does light up, things get a bit more tricky - you'd have to measure the microcontroller and/or LED driver output. Can cross that stage if/when you get there.