I recently disassembled a guitar pedal I had bought to flip. The volt adapter on it is broken so I need to replace it. I don't know if I should try cutting the original wire entirely and try to install the new wire starting at the circuit board or if I should cut the new and old wire partially and just connect them. Here are some photos.

Neither one should be difficult. For both you should have a soldering iron and solder to connect the new wire to the board (or the new wire).
If that's all you need to do, you found a great flipping opportunity. That sounds like a real easy and straightforward repair.

My concern is making the wire looks neat at least from where it is visible when it's assembled, making sure I get the proper spot on the circuit board, and making sure the wire fits as I see how the original wire had to be twisted. I have never soldered anything except for this one kit thing that had step by step instructions but I am in an electrical class so I can ask for help.