How long would it take for infastructure to fail if humans disapeared?

This question is similar to this one, which wanted to know what the world would be like 10,000 years after humans disappeared. Almost all traces of human activity would be gone from the earth - only the most corrosion resistant objects would be left.

However, I was thinking, with all the automated systems around now, how important are humans? Obviously things like bearings in wind farms and hydroelectric dams would fail, and there are parts in nuclear reactors that also need maintenance to stay running, but just how long would infrastructure like the power grid and the internet stay up and running if all humans suddenly vanished? Obviously, after the main power grid shut down, most of the internet would follow shortly after, but I'm guessing some places have pretty good backup systems in place, which would hold out for a while. I'm thinking that with all the automated systems in factories and other industrial facilities, there may be specialized machinery that could perform repairs, but may not necessarily be programed to do so.

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Then I got another crazy idea. What if, after the humans disappeared, a powerful AI took over, while the internet was still running, and was able to control robotic systems like the ones in factories, and was able to produce reliable repair bots, that could perform the maintenance that humans do? Could an AI stay operating for the foreseeable future if humans were not around to constantly repair things?

Just how important are humans to the continuation of the infrastructure and systems we have built?