How do I get this cone nut off?
It's not totally clear what's going on here, but:

If this is a cup and cone hub, your problem describes a left cone that is stuck somehow, maybe crossthreaded or the threads otherwise damaged, and you'll need to find some way of getting more purchase on it. If you have access to the right cone wrench flats, a clear way of doing that is to lock the right side locknut and cone against each other and put a cone wrench on the right cone for leverage, then unscrew the left one. If you don't have access to the right cone wrench flats, you could take the left side locknut and lock it down against the right side locknut to play the same trick. If the threads are damaged but not toast, a 1mm thread file on the axle and an M10x1 tap on the cone may help.
It looks a little like this could be a cartridge hub and the left side cone-looking bit is torqued down against the bearing inner race, whose other side is secured against a shouldered axle. If it's actually totally immobile, that would be an explanation as to why. Hubs like this usually have internal 5mm allen wrench flats on one or both axle ends to help in such spots. If not, and if you in fact need to get this part off to replace the freehub, and if the right side cone's wrench flats are inaccessible, you will probably need to do something similar to the above.