Here's a video of me doing the cube. About three minutes.
I choose a face to complete first. I often choose the face opposite white, since this makes the missing pieces of the last face easier to spot later -- in this case, the first face is blue. I complete the blue face, together with the first layer of side edges.
The centre pieces in the middle layer are trivial. I then do three of the four middle-layer edge pieces. I leave the fourth because the technique I use to place the corners in the final face disrupts the position of this fourth middle-layer edge piece.
Then I sort out the positions of the four corner pieces in the last face (white).
Then I place the fourth middle-layer piece correctly. On this occasion it happened to be correct when I'd finished the corners -- this happens in 10% of cases, but since today none of the other potential shortcuts came off it didn't save much time.
After this, I fix the orientation of the corner pieces, followed by the orientation of the sides, in the final (white) face, and finally I place each side piece in its correct position. The cube is complete.