Hi @laalman ,
Here’s a link to the [https://downloads.dell.com/manuals/all-products/esuprt_laptop/esuprt_vostro_notebook/vostro-3560_owner%27s%20manual_en-us.pdf|service manual] for your laptop.
Scroll to p.44 to view the necessary pre-requisite steps and then the procedure to remove / replace the power connector.
A replacement part is available online. Just search for ''Dell Vostro 3560 power connector'' to get results for suppliers of the part.
Hopefully this is of some help.