I don't think anybody has mentioned this yet, but make sure you have the right sized spoke wrench.
And when you get the replacement nipples make certain that they are the same size as your current ones,
having different sized nipples on the same wheel is miserable.
Remember, to be patient.
Cool off for a day, then come back to it with conviction and a healthy respect for how easy it is to round off a nipple.
One turn of one spoke at a time, then check, then move to the next spoke and repeat.
Good luck