How long does $5 soldering irons(that plug directly into mains) last?

It is daunting to buy a temperature-controlled iron. With a cheap one you don't know what temperature you are using so you don't think about it. As soon as there is a knob on the front, you have to make decisions for yourself.

But the temperature control makes it useful for more than just soldering.

It used to be that a "cheap one" was $30 and it could be used to build projects for many years. Now they are so cheap, they must be assembled in factories where nobody - even the managers - knows what the product does. (Think of Chinese workers assembling Christmas decorations. They don't know what they are for or what the English words mean.) So it is quite possible to buy a dud. Something that heats up so hot that the tip oxidizes in an hour is quite probable.

The best advice I can give is to buy whatever is cheapest at Sparkfun, Adafruit, Radio Shack or Sainsmart. If you can stretch the budget to temperature-controlled then you should.