How to fix the downspout

This isn't a downspout problem, this is a grading problem. The ground has either eroded over time or was never properly graded in the first place. Probably a combination. Either way, the end result is that water collects in the low places and only dissipates by absorbing into the ground (not good; that can cause foundation problems like liquefaction of the soil underneath a slab or P&B foundation, or weeping and frost heaves against a basement wall) or evaporating.

You're not going to fix this by extending the downspout; you may reduce the amount of water collecting, but rain will still fall naturally into this valley next to your house, or fall against the side of your house and then run down into it. You're going to have to solve the problem by creating a continuously-downhill path for this water, directing it away from your house.

This sidewalk; is it sloped in a V with the lowest point at its center? If so, it was probably built as much to direct runoff as to be a walking path, and you should investigate where the sidewalk should be sending the water (probably around to the front of the house and down the driveway) and make adjustments accordingly. If not, why does it slope? It is really old and sloping due to erosion underneath it, or did the contractor who put it in just not have his level with him that day?

Can we get some real pictures of the side of your house and this valley, with indications of where the water currently pools?

The preferred solution that I've seen is to discharge the gutter through a couple inch air gap into a catch basin that outputs into the underground drain. The air gap allows for water to drain into the yard when the catch basin freezes solid.

And while it's a bit late to change your underground drains, I prefer the solid pipe installed with a slight slope over the corrugated tubing. It's less likely to collapse over time and water doesn't have anywhere to be trapped in the tube.