That part is known as the power jack. Replacement of a power jack is not too difficult and there are quite a few reputable places that will do the repair if you do not have the tools/skills to do so.
First you can look up the model number and power jack replacement - there may be videos or "how-to" websites that explain the steps involved. You will also find listings of companies that specialize in power jack repair, normally in the $75-$100 range. The repair is not uncommon but there are a limited repair shops that will do it.