I presume you already understand what non-manifold geometry is, but if anyone reading this doesn't, there's a good explanation here.
I opened up your model, entered edit mode, pressed 2 to enter edge select mode, then went to Select > Select All By Trait > Non-manifold. This is what Blender highlighted (its the same for both wings).
I presume that the complexity of this mesh and the awkwardly placed non-manifold geometry is what is tripping up Blender's algorithm for eliminating that geometry. Let's take a closer look at what exactly going on.
The faces that are highlighted are purely flat planes. Notice how the same face is highlighted when viewing from both sides? You'll need to add thickness to those.
You'll notice that there are a couple of edges that are highlighted in the center of the wing. In these areas, the faces are coming together down to an edge, a flat edge. Once again, notice how individual edges can be viewed from both sides of the mesh. I've forgotten to deselect a couple edges that are just on the perimeter of your object, but I think you get the point.
I don't see any quick one press fixes for this... there are multiple non-manifold faces that Blender didn't even highlight, particularly at the base of the wing.
Alright, I've managed to waterproof the mesh up to the part where it needs to connect to the rest of the body. Waterproofing the mesh was simply a nightmare...Let's just say I had to take apart the mesh and reassemble it, adding in extra edges and faces where necessary.
I wasn't sure how its meant to connect to the rest of the figure (and thus how the mesh should be shaped) so I just left it open there. The bulk of the work is done though, the only open parts of the wings are shown below.