Watching a full battery replacement guide would have probably helped avoid damaging the flat cable under the battery ;) Good news is the replacement part you need is quite inexpensive, bad news is there’s quite a bit of work to do. I’d recommend doing yourself a favour and look for a cable that already has metal button supports and vibration switch mounted. It seems we don’t have a guide for a 5S but the one for 5C is just about the same and you can use that one for reference:
[guide|23004]