If you drained the machine and filled it, it should run it self through.  You just have a bit more lubrication going on now so your plugs might be starting to foul.  Keep putting fresh gas in it so it keeps leaning out the mixture of reg gas to diesel and eventaully it will all be gone.  To keep your plugs from fouling if they haven't allready, run the sit out of the machine meaning put a load on it for a while every now and again. That white smoke is probably just diesel vapor.  Stick a lighter by your exhaust and see if you can get it to light.;-) just kidding  Your little betsy will be ok here in a bit.;-) 
     It's the other way around that is bad for a motor ( gas in a diesel).  If this was a brand new machine however, you could wash your cylinder walls and your rings would never seat and she'd smoke for the rest of her welding life.

This happened to a woman's car when the gas station mistakenly put diesel in the gas tank.  She had the car towed to a mechanic and they drained the tank.  The station wound up paying for the cost of "repairs" + towing.

Relax, you won't need a new machine.
As posted above, it will just foul the plugs and provide a little more lubrication.
It could be pretty hard to start though, so after siphoning and refilling the tank, I would pull the fuel filter out, and while replacing it, bleed the fuel line a little, to get the diesel out of the lines.  Otherwise, you may have to seriously stress your starter, before you can get the line cleared out.
That, and it might be a good idea to pull your plugs, and rinse them off with carb cleaner.  The diesel should burn cleanly off -once the engine is hot-, but you may have trouble starting it with fouled plugs.