Can't disconnect plastic piece for new laptop video cable?

I got a new video cable for my laptop, but I've spent the last hour and a half unable to disconnect one little plastic piece. All it does is go to the HP light on the outside of the laptop that lights up when the laptop is on. I ended up trying to cut off one end of the plastic piece (the part on the broken cable) to try to get to a point that I can pull it apart better, but it didn't help. Looking at the new cable, I'm assuming the part that I have circled in one of the pictures is part of the non-cable piece that somehow needs to be pushed down to unsnap it from the cable piece, but even when I try that, I get nowhere.

At this point, I've destroyed several nails, and I'm probably only going to destroy the wires that I need (if I haven't already) if I continue, as when I pull and my nails slip off/chip off, I end up pinching the wires. I'm really going to be unhappy if some tiny piece of poorly-designed plastic ends up costing me hundreds of dollars just because it's stopping me from fixing my laptop...

Any ideas on how to disconnect these pieces? It seems near impossible at this point. And any idea if I even need this piece? I really don't care if I have an HP logo light up, but I don't want to leave it disconnected and end up sending too much power to the screen and frying it or something.

EDIT - Well, I went back to attempt it since no one seemed to be answering and I can't find anything else to help, and eventually the wires broke off. THEN I was able to disconnect it. Awesome. Any idea if I need it or if my laptop will function fine without being connected to the light?