Also make certain you have the correct nut (diameter, and thread pitch). In fact with something this critical I'd use a new nut regardless. Next time, instead of forcing, back the nut off right away and check to see what's causing the problem. Those threads look pretty bad, but the correct die might just save them. Then again, this is a potential safety issue - even if you can repair the threads, the strength of the axle may be compromised - not a good situation to have, especially if that's a front wheel. Maybe adding a second nut would help, and/or threadlocker. I'd defintely take that to an experienced mechanic to check out before using - it's too difficult to tell if that's safe to use based on your picture alone.

Make sure to practice using the die, or a die like it, on a spare bolt first. They're not hard to use once you have experience, but it's also an easy task to screw up. I would hold the axle straight up in an axle vice and keep the die handle as straight as possible. Defintely use cutting fluid.

In fact, I'd strongly consider letting an experienced mechanic with the right tools do the job. It wouldn't take long, and would probably be the least hassle. You could still learn how to use taps and dies later... on something that isn't protecting your safety.