Good idea. Things are not computing here at all. 10 hours of welding (60 amps at 220 V) at an exorbitant $0.25/KWH would only raise your electric bill about $33. That 10 hours would be actual arc time. That's many lineal feet of weld. If you weld at one foot per minute, it's 600 feet of weld. A welder that is switched on but not actually welding doesn't draw much power; if it did, it'd get very hot very quickly.
An unsuspected load like an iced-up dehumidifier can raise the bill a lot, but that doesn't jibe with the spike you report when you do a little welding.
I would definitely check the electric meter. Turn off all loads you can spare, read the meter, read it again after an hour. Then turn on a known load, perhaps a 1200 watt hotplate or whatever, do the meter drill again and then do the math.
I have never had an electric bill of more than $150.