[villagetinker]

OP, I like the idea of covering the holes on the INSIDE of the lanai and the use of sponges. So IMHO, try doing both, then have your better half go to one side of the lanai and you to the other side and one of you take a hose and spray the sliders while the other looks to see where and how the water is coming in.
This should give you a real good idea of how to fix. If you do not see water in the lanai, but do see it in the aluminum channels, then the drain holes are incorrect. add new drain holes, try applying water again, and if it drains out, then make arrangements to permanently seal the holes into the lanai.