It appears that the +ve of the LED on the half of the board with the black wire was connected to the -ve of the the LED on the half with the red wire, and that that connection is the only broke one. You could just connect to the round pads (+ on the half with the black wire and - on the half with the red wire) and not have to worry about scraping the solder resist off.
Using something as a splint and glueing the halves to give mechanical strength is a good idea. The original board was both a mechanical and an electrical connector, like most circuit boards, which saved assembly time. However, you do not have to use the broken tracks on the board if you don't want to. It will be much easier to solder to the existing pads.