How to fix a tap where the handle has fallen off
From my point of view (image is very small and poorly illuminated) you face a hexagonal ending of the shaft, when looking at the top of the tap. My first try was to get a wrench socket, which fits the hexagonal stub. It may be difficult to find a matching wrench socket small enough to fit into the housing of the tap, but you could take a cheap one and grind the outer diameter down until it fits. then you screw it down with a matching screw and insert a short t-handle. Alternatively you may fix a valve handle to it via a sqare axis reaching from your wrench socket into the handle. 

Update
there's another way to make place for a wrench socket (assuming there's a wrench socket fitting the hexagon, but not the inner diameter). You could saw off the upper rim of the tap surrounding the hexagonal stub of the shaft, so you can place the wrench socket on it, without having to grind its outer diameter down, which is undoubtedly the harder work to do. The wrench socket — again — can be fixated by a matching screw and some washers on the tap to prevent it from popping out.