The red button is part of what is called a thermal circuit breaker. They open circuit when there is a current overload and pushing the red button is designed reset the breaker back to the closed condition. They are used in gear like this because they are low cost. But along with that they have a limited life time and limited number of trip cycles before they fail.
You could check as to where that cord reel was manufactured and if it has a legitimate safety agency listing. There are many products manufactured in certain countries where the quality of components is really low and fake safety agency marks are applied to the product labels. If your reel is one of that type it probably failed due to a crappy thermal breaker.
Even if the thermal breaker is a better quality unit it may simply have been severely over stressed by the user trying to run a welder on an extension cord like that. There was probably no attention paid to the ratings of the cord and the power requirements for the welder.
It may be possible to replace that thermal breaker if it is a standard component. If it is a custom or unknown component then your fixup job could be a bit more complicated. There are panel mount type thermal breakers that you may want to research to see if you can find something that would work as a replacement. But do be aware that as soon as you start to repair a low cost device like this, if the device does indeed have legitimate safety mark, that you would have to use exact replacement parts to maintain the certified safety rating for the device. Using cheap replacement parts may very well not be safe at all, especially if they are junk from certain countries.