I have always had good luck with the cobalt coated drill bits when drilling through tough metal. Don't use the hammer drill, it will chip the bit. Also get some cutting fluid, WD will work in a pinch. Don't over heat the bit, keep plenty of fluid on it. If you overheat the bit, it will take the temper out of it and then it is almost useless.

 

If you drill it, a drill press would make the job much easier. For the price of that axle, I would just go new...

 

Slow RPM's and lots of push. Yes, a drill press would work, but the axle is on the quad and I don't think he can remove it. Or at least doesn't want to. I'd try to find a pry point behind the drill. Maybe put the drill vertical and pry between the frame and the drill with a 2 x 4 or something. You may need to go through with a small drill first. Then move to the 1/4".

 

Sometimes you can put more money and effort into rigging something then fixing it right - this is likely one of those times.

Ive done this all before on machines I was "just trying to get by" with - nothing has lasted more than a few days with the exception of one....which lasted 6 months.

Get a good used axles, swap it out and just be done with it.

The repair I got to last the longest (the above mentioned 6 months) was JB Weld. I caked it on the axle, slid the wheel on and let it setup good - left it like 2 days so I was sure it was strong. It really did last 6 months before it broke again.