If you can find a replacement cord, any splice will need insulation for each conductor inside, which make a surprisingly large bulge in the cord where the splice was made. This becomes the new site where stresses congregate, and reduces the life of the repaired cord.
The power brick was never intended to be opened, although you can saw out the glue along the seams and pry the halves apart. The wires attach to interior points on the circuit boards inside, which are not immediately accessible for soldering unless you do further dis-assembly of the electronic components inside.
I have done such repairs, and found the results were not very good. I recommend you buy a replacement power adapter.