If you have a pool service company, they would be my first suggestion. The pool should not drain towards the lanai, and the pool drain should be at least a few inches lower than the lanai floor. I would also look where the drains exit to the lawn, you should be able to snake these and remove any built-up debris. I am guessing on some of this as I have no idea of how your lanai and pool are setup.
You might be able to color some water with food coloring and pour it is suspected area to see if it comes in the lanai.
On a side note when our lanai was enclosed the installer were very careful to seal the bottom rails so water would not come in.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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