Disassembling AC adapter
I would not open the adapter at all.  Cut the cord a few inches from the adapter, and splice on the cord that you want.  You can do the same thing at the other end if you want to continue using the existing plug.

This has the advantage that you aren't messing with, and possibly defeating, any of the safety systems.  All the work you are doing is on the low voltage and on the safe side of the isolation barrier.