The switches are two part affairs, made out of plastic. The broken switch has literally, fallen apart, but I'll get you some pictures of it. Mind you, this is from a truck, but it looks to be EXACTLY the same as my Suburban:
The first link is of the top 1/2 of the broken switch; the dangling black wire with a loop is the third wire that goes into the switch. It sits on the pin in the middle of the top half:
These are the components of the switch:
From right to left, of the picture JUST above: Round black button, bottom case of switch, as it mounts in the truck, internal button piece, spring that fits inside, additional button piece.
Picture above: The button pseudo assembled (without the spring;l it would not stay together with it, for these pictures.
That black piece comes out; you can see it in the second picture from the top. It is supposed to sit over the black wire whose loop is over the the center post. Then the white part of the button sits on that, then the spring, then the black part of the button, then the actual part of the button that users push.
Put it all together and it looks something like this:
It is tricky to pull it out of the case, if the switch is intact; basically, you gently pull the whole thing toward the prong side.
The biggest problem with these switches is that the plastic gets brittle as it ages, which causes the two case halves to split open. And that is when the switch dies.
I have seen one or two write-ups where the case sides were glued back together, but the nature of how the switch works leads to them breaking again within months of the repair.