Due to a power failure, I had to change almost all the water in my son's tank while he is away at college. I am having to do a cycle with fish and have never done it before.
The ammonia spiked, then dropped to 0 and is still there. The nitrite spiked and I have done 1/3 water changes daily, keeping salt at 1 tbsp. per 5 gal. The nitrite is now at .125ppm. Is this low enough to try leaving for the bacteria to use up, provided it isn't higher tomorrow? or should I keep up the daily water changes until nitrites are at 0? We have no test kit for nitrates, I will get one next time I'm out. Does it sound like I'm doing this right?

Hey, well done - it sounds like you are doing everything right. And welcome to Koko's

I would continue with 10% - 20% changes daily until you get a zero for nitrites, as although 0.125 seems really low, even at a small concentration nitrite can make fish sick (or worse still, kill a sensitive fish).

The bacteria will eventually be able to cope, but until then - a little help via water changes will keep everything good.

Okay, thanks.
I was just worried that if I removed all the nitrites, the nitrite eating bacteria wouldn't be able to multiply enough to handle all the fish waste, so nitrites could spike again. If that makes any sense at all.

It makes sense, but you will never remove all of the nitrites unless you took out 100% of the water. Even removing 50% of the water you will still have half of the nitrites left in the tank.