Yesterday I noticed that it took a considerable larger amount of force to turn the steering wheel. It was stiff at low speed (compared to the usual feeling), turning onto streets and parking was not easy but manageable. I did not know what the problem was, so I was just thinking of getting my car checked the next day.
Then I washed my car and noticed that one of the front tires was
completely (EDIT: very, I've never seen it that flat but I don't know if that's "completely") flat. I am pretty sure the tire looked ok when I left my car 12 hours ago, so I figure it must have gotten flat in the past 12 hours, and probably contributed to the stiff steering wheel. I drove to a repair shop, where they said I probably hit a nail or something. They removed the tire, patched it somehow (I saw the technician removed something with a metal tool then applied some kind of glue, but I did not have a clear view to see exactly what he did), pumped it and put it back. The steering feels alright afterwards. They charged me for around USD 10 equivalent (I am not in the US).
My question is, how bad is the damage to the tire? Maximum speed was 120 km/h, and I drove around 40~45 km before I got it fixed. Should I expect a significantly shorter tire lifespan? Should I replace all tires at the next service interval?
Are there any other maintenance steps I should take after this simple fix? E.g. should I go to a shop and ask for the tire balance be checked?