First let me say that underground French drains do not work very well in northern Illinois unless they are very large. The problem is that we have too much clay in northern Illinois. I suggest only building French drains that end in a ground level decorative gravel fake stream or pond. Open Gravel areas will allow more evaporation than underground gravel pits.
Start by trenching from the water problem area to desired location for your gravel area, Making sure to surround the drain tile with river rock as you go. The gravel area should be dug out about a 2 foot depression. Then spiral the drain tile in the depression to fill the area and bury with river rock. Top off the area with decorative rock of your choice. I like Mexican pebbles.
Then go back and top off the trench line with top soil and sod. Use a compactor to seal the sod to match the ground level. Then finish your gravel area off with the water garden plants to decorate the area and make it look more natural.
Here is a list of water plants that I like to use, SporobolusPanicum, Miscanthus purpurescens, Cornus sericea, Cornus racemosa, Betula nigra, Acer rubrum, Acer saccharum, Aronia, Clethra, Eupatorium, Dicentra, Heliopsis, Ilex glabra, Ilex verticillata, Hosta, Iris versicolor, Nyssa, Hemerocallis, Polemonium, Quercus bicolor, Sisyrinchium, Solidago, Viburnum, dentatums, Viburnum prunifolium.