I just had to complete this same repair. I got a replacement contact board off eBay (Charging Charger Connector With Adhesive For Samsung Gear 2 SM-R380 Neo SM-R381) and dropped it in. Hindsight being 20/20 and all I should have taken a ton of pics to go along with but oh well. I was originally planning on replacing the entire rear cover but ultimately decided on just replacing the contact board.
Anyways, a few things to note (reference pic here http://i.imgur.com/Z86H9Cw.jpg, don't mind the destroyed board in the center - I tried lifting the old one out and the PCB tore away from the contact assembly)
1) When buying a new contact board, get the ones with the adhesive already on the gasket - it's protected by the little blue tab in the center there. Some new covers might have the adhesive already in place so don't double stack it.
2) The easiest way to get the old contact board out is to take a flat screwdriver and press on the contacts from the outside. With a little bit of pressure the board will come up clean. The Spudger worked great for this.
3) Once all of the old adhesive is cleaned off, hold the new contact board by the little tab with some tweezers, then remove the adhesive cover and drop the new board into place. Tiny bit of pressure to set the adhesive and then you're good to go!
Hope this helps!