Honestly, if you're constantly breaking power jack, do you think soldering the power cord straight in is going to help? The same force that breaks the jack is going to yank on the power cord . . .
First step would be to look inside and see if you are capable of making the solder connection. If you are, then you can plug in (what's left of) the power cord and take a voltage reading across the two points you would solder to. That will identify + and ground. Cut the jack off the power brick and use your mm on those two wires to identify + and ground. Solder like together.