One problem PiKnife had a lot was that the stepper motors would sometimes stick and jitter. Eventually, I noticed that pressing on the input and output wires plugged into the ribbon cable going to the Raspberry Pi would cause this problem, so I deduced that the issue must stem from bad connections. To fix the connections, I wrote down where everything was in the ribbon cable and then used male-to-female breadboard jumpers to wire each input and output directly into the Pi's GPIO header. This eliminated the uncertainty of plugging wires directly into a ribbon cable.
Amazingly, that fixed the problem entirely! I set a new piece of paper on the laser cutter, pressed enter to begin the program, and didn't have an issue again! While it took five and a half hours to complete the design, it showed the true capability of PiKnife. It can cut extremely complex shapes for a long duration without problems.