How to replace game card slot

Check if the pins are broken or bent. If they are bent, you might be able to use a toothpick to fix your problem.

If the pins are broken, then you will need to probably solder a new game card reader/port/socket. This is not an easy fix since you will need to use a heat gun to SMD solder it on next to very sensitive components. Here is a link for a replacement housing.

You could potentially get away by soldering wires to each one of the exterior pins of the old housing to a new housing that would dangle out on the back using wires. It would be a fragile but functional fix that would only use a soldering iron.