A new light is probably in order.
The green circuit board looks to have surface mount components on it, and replacing these would generally be beyond DIY at home with a soldering iron.
The switch is the larger square item labelled S1.
Sometimes the outside part of the switch is a flexible rubber cover, with a harder rubber cylinder nested inside that actually prods the switch on the circuit board. If the cylinder went walkabout with the original cover, you should be able to turn the light on and off by prodding the switch with (eg) the eraser end of a pencil. If that's the case, it's a matter of finding something that will fit in the new switch cover you've got and is long enough to reach the actual switch.
The (more likely) alternative is that the switch on the circuit board had an upper part (now missing) that connected the visible contacts when prodded. In that case, prodding what's visible with the back end of a pencil won't do anything.
You may find that shorting the centre & upper, or centre & lower, contacts with a paper clip will turn the light on/off, but I can't imagine that would be acceptable for long.
/*just guessing*/