Replacement wall cord for power brick?

I recently acquired an old HP Compaq NX 9600. When I got it home and tested it, it booted right up and ran for about 15 minutes before it gave the low battery signal and powered off. I took out the power adapter and plugged it in, but the computer would not start. I finally noticed that the green light on the power brick would come on and immediately dim and go out within about 3 seconds. I then realized that the ground prong on the wall side of the power cord was completely gone. The adapter is set up such that on one end of the brick is a simple wall cord section that can be unplugged from the brick. The other side is fused with the power brick and connects to the laptop. (bear with me if that is stating the obvious, I am somewhat new to the "mobile world")

I noticed that the connection between the damaged wall cord segment and the brick was the exact same shape as the wall cord used for one of my spare LCD computer monitors. I tried connecting it and sure enough the light came on, stayed on, and the computer booted up. My question is, are there any compatibility issues between the wall cord portion of the power supply and the brick/computer? I know that the power brick has to meet certain amps and wattages, but does the cord that plugs it into the wall matter? Is there any kind of safety concern I should know about, or is this portion universal and using the monitor power cord be fine as long as it fits? Thank you so much for any help you can give me. I don't want to order an entirely new power supply, but I am planning on ordering a second monitor cable since they are easy to find and appear to work. I just want to be sure there is nothing else I need to consider.