Just do it yourself, 220 just hurts for a minute.
You've got room in the panel for a breaker go to HD pick up a 50 amp breaker, make sure it;s the same brand that you already have. Punch a hole in the back of the panel for the wires get a 4x4 metal box, a 220 recpticle wire it the same way your A/C is wired and yer welding. If that Romex is waht you were planning on using it ain't big enough. You should be running like 8 guage.
My advice is while you're at home depot anyway buy 50' of 50amp 4 wire cord and make yourself an extension cord for the welder so you can weld anywhere in the driveway. My extension cord gets me to the middle of the street for welding on trailers and stuff that won't fit in the driveway.
Make sure when you get it working that any kids or neighbors that come over and need stuff welded that you do it with a smile it has kept my complaints to a bare minimum and kept the kids from effing with my stuff when the get older.