I know that Inspector Gadget has a dizzying amount of gadgets, but does anyone know how many gadgets Inspector Gadget actually has?
We don't know how many gadgets he has and we aren't meant to know as its not important to the plot. What is important to know is that he has, in your words, a dizzying amount of gadgets, all available to him literally at his fingertips whenever he needs them.
His seemingly unlimited supply of gadgets is a plot device and gag, not un-similar to other famous super-heroes such as Batman's utility belt (go go gadget shark rpllant anyone?), or Iron Man's suit. We don't care as an audience where or how so many gadgets are so readily available to these characters because the only thing we need to know is that they are technologically advanced and resourceful.