I've replaced a few power supplies in monitors. This is perhaps one of the more common "monitor deaths". Mostly it looks like electrolytic capacitor failures. When those capacitors fail sometimes they short, but I've never seen one melt the cable; there are fuses in these, and so I might open the monitor up and see what other obvious damage there is. For a cable which melted I'm more likely to just discard/recycle the monitor, but if is something really simple, e.g., mounting screws coming loose and wires directly touching, it might be that the power unit itself is ok, but you're taking a big risk any time you buy replacment components.
Many monitors have such common capacitor failures that aftermarket people sell the capacitors for a rather high markup price; finding some parts at "ordinary" component prices helps a lot if and only if you are able to troubleshoot and repair individual components. There are also often aftermarket whole power supplies for particular monitors, but these are often "iffy". The replacement supplies themselves often sit on shelves for over a decade, which is why they are available, and in that case, the electrolytics can be close to failure just from sitting there that long (the life expectancy of such a supply is shortened). If you happen to have a complete replacement supply for a good price it can be worthwhile.
In the end though, unless it is a particularly noteworthy monitor and it is perhaps worth risking losing more money on it, I would be tempted to dump that monitor. That melted power cable is particularly risky, and much more might have died than just the cable.