The best thing to do is not to have it drain under the patio -- when you change the moisure content of soil, it can compact differently, or wash out, both of which could cause the problems you're seeing. If you can, change the drain there so you're using non-perforated pipe, and run it out further to a drainage field past the bricked area.
The problem is, you want to compact the soil for the patio, which will reduce the air voids in the soil, making it less permeable and not drain as well, making the french drain less effective.
As it's bricked, if you slope it away from the house, you'll reduce the amount of water absorbed in that area, which should hopefully eliminate the need for a french drain to collect the water to take down to a drainage area.
If you want to get the best compaction of your soil, dig some if it out, then compact the soil, then add a few inches back in, compact again, then repeat until you're out of soil. Then repeat for the base pack, check it for level and then start laying your pavers.
Here is a pretty quick explanation.
Starting from where the water will enter the drain and ending where the water will flow out, dig a ditch that's about 10 inches deep and 6 inches wide. Keep in mind that you will need to dig deeper if your starting point slopes upward.
Lay about 2 inches of rock into the trench before placing the pipe lengthwise into the ditch.
Cover the pipe with another layer of rocks, again about 2 inches.
Fill in the remaining space with soil and then plant some grass.
Here is a pretty good video, which also shows it's a good idea to add a clean-out.
They also say you should not use flexible corrugated perforated plastic drain pipe, because it cannot be cleaned like rigid pvc can be.
Best Answer
These materiasl will work. Lining the trench with landscape fabric and wrapping it over the top (of the rubble) will keep silt from clogging your drain. For higher volume flow, I'd add a perforated pipe at the bottom, but it's surely overkill for just a sink.
Definitely make your rubble somewhat uniform, for best support for the overburden (assuming you are burying the drain).