There are no other option but to change the iPhone 5 charger port dock connector flex cable, which already comes with head phone audio jack. You get one for less than $9 on Amazon with free shipping. Here is a link on Amazon: http://www.amazon.com/iPhone-Headphone-Replacement-Instructions-Included/dp/B00LSYZT12/ref=pd_sbs_468_6?ie=UTF8&refRID=0N3JB9TTEQWJ9Z87HQAR
Installation is very easy and all screw drivers are usually included with the part. Just make sure to pick the proper color (white or black) for your iPhone.