I think the HDMI cable is broken

First, you might want to try a different HDMI cable. If it works with another cable, you've probably got a faulty cable. I don't remember if that particular console has multiple output ports on it, but if it does, you could also try using a different type of output (i.e., AV, Component, etc.) just to make sure it's not the GPU or something like that. If a different cable doesn't work, check the port and make sure there isn't anything stuck in it. I've heard of that sort of thing causing issues with video output. If you can't find anything or cleared out an obstruction, it might just be a faulty port. This probably requires soldering if you don't want to replace a whole circuit board, and I'm assuming you don't want to do that, but if you know how to solder, that shouldn't be too big of a problem. If not, you'll either have to replace whatever circuit board the port happens to be on (I'd guess it would be on the motherboard, so you'd also have to replace the optical drive-I think they might be paired), learn to solder (It's easier than it sounds, but given that you'd be soldering an HDMI port, I would probably practice on something else if you don't already know how to solder), or find a repair shop and see if they'd do it.