I can see that the magsafe port needs to be replaced. Your picture shows the damage quite clearly. Your MacBook has been dropped (see the crack radiating down from that opening?), and the cracks on the Magsafe connector likely mean that the magnet has been knocked loose, and is likely attached to something else magnetic, maybe one of the internal speakers. Take your Macbook to an authorized service shop, or and Apple retail store for service. They will be able to give you a quote for the repair - and an experienced tech can make sure that the magnet inside is not loose to cause further damage.
They might even replace that missing bottom case cover (worth asking for that!) Sometimes they do that at no charge, but I'm not sure about that.
If you still have your old charger cable, can you get the charging power light to come on, if you hold the charger in position? If it is a right-angle connector, and not the older straight connector, it would be fairly simple to tape the connector in place, so you can charge the battery, and use your MacBook.