Is it gone after 24 hours? Builders around here require 48 hours. The heavy clay soil drains slow around here. This happens on every step lot around here. A friend of mine has this same exact problem.
Based on my experience, here's what I'd do:
Get 3 or 4 drainage basins with grates for 4" pipe. Square, round, whatever you prefer. Get 3 or 4 wyes as well. At the other end where the water will drain, you can use a popup emitter another drainage basin or just leave the end of the pipe open if you have enough fall to cut it at an angle to match the ground.
Dig a trench about a foot wide and about a 18" deep from the start of the wet area all the way to a low spot that can drain naturally. For you, it looks like that's your front yard.
Dig holes for the basins a foot or so to the side of the main line (The first one at the end can be inline, however). Dig a trench from the basin hole to the main line at an angle that matched the wye adapter. The reason to do this is because I don't like to have my basins inline with the main drainage pipe because the water will flow through each basin on it's way through the pipe causing the basins to drain slower during heavy rains plus it will take the sediment from the basin into the pipe and clog it up.
Taper the ground out a few feet from the basin holes so the water will drain toward them.
Place your basins in the basin holes and make sure you have some fall from the basin hole to the main line. I like to have enough fall to where the top of the main pipe is below the bottom of the basin pipe hole. This may require digging the main line a little deeper than a foot.
Once everything is the correct depth, lay your main line in the hole making sure to cut it and and install the wye adapters and connect the basins to the whe adapters with short sections of pipe. Place piles of dirt on the pipe to help hold it in place as you go.
Cover the pipe with dirt. Don't pack too hard to where you crush the pipe.
Final grade and reinstall the sod.
Here's a little text diagram of what I'm trying to say.
O===================================================O Low End
// // //
O O O
You can also run your gutters into this system if you want.
If you don't want basins, you can get perforated corrugated pipe pre-wrapped in a sock and put it in an 18" deep trench, back fill with very coarse sand, cover with well draining top soil then cover with sod. There are many different variations of this. There's a drainage system called akwadrain that I've used successfully. You can wrap the hole with drainage fabric, install a pipe then rock then wrap with the fabric then cover with soil then sod, etc...
I prefer the pipe with basins if it's just surface water from rain and sprinklers causing the problem like in your case as it will almost instantly remove all the water.
If you have saturated soil like me, I use perforated pipe with no basins.
