The "Lab Series" Audio Technica's were good cartridges and came out in the mid-80's just before or during the introduction of the CD. Some of them are near the top of the line of AT's cartridge line.
I'm not personally aware of the LS580, but the LS500 is a very good cartridge.
The $45 stylus replacement you mention is an off-brand piece of junk. It will fit, but not do the cartridge justice. It's not worth $45, that's for sure. However, the stylus you really need is an Audio Technica brand for the LS580 and you'll not only have a hard time finding one, it will be very expensive -- about 3 times more than the junk one you see on the web. However, I've looked and I doubt you'll find one.
Trust me, you don't want to spend $45 on that stylus. Your cartridge body is worth using, but not with such an inferior stylus it would be wasting your money.
You are better off giving up on that cartridge and getting a new AT440ML. It's a very good cartridge and doesn't cost much more than the $45 that the junk replacement stylus costs.
It is possible that the AT440ML stylus would fit the LS580. I know of other AT carts that this is true for.