[Highland Park] Help! Clogged outside drain

Hello! The drain from our roof downspout is clogged, we discovered it when we cut the downspout to attach a rain barrel. My dad and I have dug about 3 feet of dirt and muck out of it. We snaked it with a toilet snake, plunged it, and used a string mop as a plunger to get closer to the source. Nothing is working. Home Depot wants $50 to rent a large snake but I don't know what it needs and don't want to go through the time and expense of hauling it home only to find that it won't work. Please email me if you know of a good, reasonable plumber who may be able to help. Thank you! Sherri