How do I remove a destroyed lug nut?

Well, the only thing I can suggest is to drill it out with a sharp drill, but for that to work there will need to be a drill-guide so that the wheel does not get damaged (anymore...)

Getting a nut off in this case can be done, how you would attack it depends on access to it. If the nut sticks out you could use a nut breaker to crack it off, or get a big pair of locking pliers, tightening the damn thing down as much as you can.

If you can't get a pliers or nut breaker on then I'd try to use a rotary tool and a metal cutting disk to cut a deep groove on the top of the nut, then use a big, fat flathead screwdriver on it. Be careful not to cut into your alloy doing it though, and protect the alloy from the sparks with some tape.

Last you can also buy special socket tools which wind on.

Before you do anything though spray the nut and the area around it liberally with penetrating fluid like WD40. Do that several times and give it an hour to soak in. Works wonders.

For stubborn nuts , heat the nut with a torch, then move the flame away and immediately apply WD-40 or any other penetrating oil against the heated bolt threads. The quick change from high heat to to the cool oil will cause the nut to retract and expand, allowing the penetrating oil deeper into the threads to create a slippery surface. You can do this multiple times. Just make sure there is nothing flammable around when torching. After doing this a couple times, use a breaker bar to pull it free.

If that fails, I would then use an angle grinder to cut a groove in the nut, and use a flathead bit on an impact to get it out.

Good luck!

Torch it. Melt the lug to slag and remove the wheel. Replace the stud yourself or drive to a mechanic with 4/5 lugs on and have them do it.

Get a socket you don't mind ruining and weld it to the nut. Then use a pipe on the end of the socket wrench for some leverage. As messed up as that lug already is, use a decent quality socket wrench or that thing could break under the force (I've done that a couple of times). If you don't have access to a welder you can try some JB Weld (lots of places sell it). Use the JB Weld as a "last resort" though, because if it doesn't hold the socket there may be residue on the nut itself making things even tougher.

Just know that you are probably going to lose the bolt (wheel stud) as a result of this. It isn't a huge deal: assuming nothing else is damaged, it can be replaced without too much difficulty.

The safe way to do this is put all tires on as well as you can, as many lugs as you can, and get a tow to a shop. Let them do the socket welding trick. Also, please egg your neighbors' house.

Get the kind of cutting wheel that fits on a drill motor so you can cut almost flush with the wheel. Cut off both the nut and the lug. Then you can drive with the other four lugs/nuts to a place that can replace the cut-off lug.

I bought a car that had wheel locks. Basically weird shaped lugs with special sockets to remove them. Of course the previous owner lost the unlock sockets.

On 1 wheel I put a socket over it and hit it with a sledgehammer. The socket split. I tried again with thicker sockets. It worked...eventually.

The other 3 I took a grinder and cut a flat edge on 2 sides of the lug nut until I could slip a good size (I want to say 11/16) wrench on it. That seemed easiest, the other 3 locks were off in about 20 mins.

The hard part is that if it's stuck hard enough that it didn't come off before, the various normal methods for removing a stuck bolt may not get enough grip. Also it looks like things are bent enough that maybe it's touching the hub rim...

But, if the other methods fail, short of buzzing the whole thing off with a grinder and then drilling out what few threads remain and replacing the stud, you might be able to split the nut. Take a dremel-type rotary tool and carefully slice down either side of the nut but not so deeply that you damage the threads, then stick a large cold-chisel (metal-cutting chisel) in the notch and give it a couple good whacks with a large hammer. With luck you can break the nut in half without too much damage to the stud.

Otherwise, buzz it off, drill it out, replace the stud. Should take less than three hours if you have the tools and are good with your hands. Put some PTFE tape on the threads when you put the nuts back on to keep them from rusting in place next time since your lug nuts don't have caps.

Im going to try a hole saw cutting enough of the outer portion of the nut to break it out (hopefully) using a hole saw small enough to pass threw the hole in the rim. Basically cutting the threads off. If that doesn't work I'll hole saw right through the rim big enough so the nut will fit through. 4 out of 5 still there is good enough for me. ;)