How should I install a french drain in this situation, and will it help?
The water table around my house is very high with a lot of surface water due to clay subsoil. My experience of 25 years in my current house (boat?) is adding a french drain around the house is a patch to a larger yard grading problem.  I've done this and eventually they clog and need maintenance.  Over time I've regraded most of my yard so water flows away from the foundation.  Gravity never fails and never clogs!  My guess is your patio is sloped incorrectly towards your house.  Short of replacing the patio, a french drain is the only alternative I can think of.  They work when installed correctly.

I just replaced my patio this summer.  We had a surveyor transit, built up the area with dirt/aggregate and put in a slightly steeper pitch. The one small part slopes towards a french drain to my downspout.  I was afraid the water would run into my neighbors yard and I don't need those problems!  The larger section pitches towards a special garden area that was dug out and filled with mulch and water loving plants.  It gets very soggy!  Again, my approach is to slope the water away from my foundation and then do something with it there.  Through hard knocks I've learned solving a water problem at the foundation is the least desirable place to address the problem.  Unfortunately for many, prior mistakes make it the only alternative.