If you're planning on continuing using the laptop, I'd recommend non-destructively removing the monitor cable rather than cutting it completely. Remove the battery, unplug the power, and unscrew the little screws on the bottom.
However, you might want to ask yourself what do you mean by a "good" laptop if it's literally falling apart, has no screen, and is being used as an underpowered desktop. You'll probably end up replacing it anyway when the battery or power supply dies. You should seriously consider purchasing a new (or used but in one piece) laptop and using that instead. If the hard drive contains anything important, it can be transferred over.