I was thinking about the deep drains, but I don't think they'd give you enough volume to make a difference. You've got more than 50 gallons of puddle there.
When I was a kid our driveway was poorly designed and only about 1/4 of it drained to the street and the rest drained back towards the house and made a huge puddle by our side yard gate (not as big as your puddle though).
I was probably 15 years old and had been taking music lessons for years and really wanted a new instrument, but it was more than I could afford. My dad told me he's pay half if I did a job for him. He brought home a 55 gallon drum with the bottom cut out of it and handed me a shovel. It took me days to dig a hole almost as deep as I was tall. Then we sunk the drum and surrounded it with gravel and topped it with a large drain fixture and grate to funnel the water into the drum.
It worked great for us, but I don't think we ever had more than 55 gallons of water to dispose of.
It's a whole lot of work and you'd need multiple drums. I think the only reason it worked for us is that we live in an arid climate and the drum was always empty until it rained. It's pretty green where you are though, so I don't think you'd start off with an empty receptacle like we did. You'd have to sink too many drums down there to make any difference.
