How much water needed to test sump pump?
Hard to be sure from that picture, but yes, it's probably an enclosed/caged float switch of some sort, and it should definitely take more than an inch or two to turn it on. Why? only pumping an inch or two is bad for the lifetime of the pump, which does most of its "wearing out" in the first few seconds it's switched on, so you want enough water that it's worth pumping for a while. If you get above the top of the pump itself (black parts) and it fails to turn on, something might be wrong with it, though before that point you might want to start with checking the outlet for power. 

If the pump IS broken, the water will, barring a water influx event, tend to go back down where it started within minutes to hours. Since the pump is not turning on and the pit is not overflowing, you don't appear to have such an event going on right now.