I recently found that the drainage system for the gutter runoff is completely clogged. It has roots running through it. I dug up the drain 15 feet (5 yards) away from the house and it is still clogged in both directions. I tried clearing it with a hand held plumber's snake and a pressure washer but I did not have much luck. How do I clean this out?
You could try hiring an electric eel
or get a plumber with a jetter
But either of these options may still not work it the pipe is broken or snapped and the two sections of pipe are now ofset from one another. Even if these options do work to clear the pipe you still have to remember that the cause of the root problem is still there, cracked or broken joints or pipe, which will require ongoing treatment eg. running the eel through once a year or similar chemical treatment.
The other option is to get the shovel out or hire a mini digger and replace the pipe with a pvc drain
I would abandoned it and dig a new pipe system. I would try and get a 20' piece of the light green PVC pipe. The roots are looking for water, this way the roots will only try to enter every 20'. I would not use any black flexible pipe.
Another thing to consider is run a piece of pipe or downspout on top of the surface. That is u less it can be seen or it is high traffic area that kids would step on. You want to get the roof water away from your foundation and keep it running away down a slope.