By broken do you mean the plastic piece is broken or the electrical switch under it doesn't do anything? If the switch button is loose it might be possible that either problem is present, but I'd imagine that just broken plastic would not stop the electrical switch under it from working unless it's wedged in somehow and not allowing normal travel of the switch.
Switch button plastic, pull it from a junkyard donor vehicle or buy the whole module on ebay/etc. By pull I mean the junkyard might want to sell the entire module but you may be able to swap just the faceplate if that's easier.
Electrical switch under it, you'd have to disassemble it enough to get a good look at whether you can source a replacement and desolder old, solder new switch on. It might take some digging but an electronics supply house like Digikey.com "probably" has something that it's drop in (solder in) replacement or could be made to work depending on the clearances.
If the switch still seems to have the same engagement movement/feel, it is possible that the switch is fine and just needs a broken solder joint to the PCB reflowed, or if a solder pad is torn up from the PCB, to cement it down and/or solder a jumper wire from the switch to the next secure area of that circuit trace, then if the switch needs a mechanical fortification to the PCB to keep it still, put a bead of epoxy around it, making sure it doesn't interfere with the switch plunger movement nor mounting the faceplate back on. There are other more advanced ways to resecure a switch instead of epoxy, like creating a new solder pad out of copper foil, but it's probably beyond the scope of a single switch repair.
It may need opened up to see what's going on then if you have questions, post good (well lit, high-resolution) pictures on a hosting site like imgur.com
