There are two problems:
Your geometries are multipart geometries. Convert them to single part geometries.
You must have a continuous line throught the whole polygon: splitting the polygon with the blue line works, but not with the red lines. The lines are treated separately for the split if you have separate lines. Connect them if you want to use them together to split.
Consider one of the red lines (the upper one): if you have only a line reaching from the outside to the hole, you won't split the polygon, because of what is to the left of the line still is contiguous with what is to the right of the line. So this operation will not result in any split. Now add the second red line (the lower one) and repeat that: still no split.
When you connect the two red lines inside the hole, the polygon will split correctly.
Blue line will split the polygon, red lines will not: