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That's exact;y the type of drain I unblocked last week. Had been blocked nearly solid with sediment and crap from years of neglect and probably from when we got our roof redone. I was able to get the cover off and used an attachment for my pressure washer (about £20 off ebay). Typical drain rods might not be flexible enough to get around a tight corner (the first set I bought were too stout). To be honest, even a fairly stiff garden hose would have worked for me given enough patience. The attachement I got I used to poke at the blockage to break it up and the pressurised water helped to flush it out of the way.
I usually use Rentokill, they usually charge a flat fee and get the job done. Only because there are so many cowboys in the drain clearing business.
It's worth carrying out a bit of investigation yourself if you can. Maybe even poke around a bit. What are the symptoms?