When problems arise on an extended network it is almost always the extenders that are creating the issues for several different reasons. Often, if an extender connects using WiFi, it's location is not correct, or there are too many obstructions in the signal path between the WiFi devices, and or other interference from nearby wireless networks. I'm not a fan at all of extending using a WiFi connection, but sometimes there are no choices.
When an extender connects to the network using a wired Ethernet connection, users frequently apply the wrong setting to the Express and set it up to "extend a wireless network". That would seem like the logical setting to use, but surprisingly is the wrong setting to use. When the wrong setting is used, it can slow down the entire network tremendously.
The correct setting is "extend using Ethernet".
We will first work on the Express that connects to the network using Ethernet. Please leave the other Express powered off for now until we know that the wired Express is working OK.
Power up the Express that connects to the LAN using an Ethernet and let it run for a minute
Then, hold in the reset button for 7-8 seconds and release and allow a full minute for the Express to restart to a slow, blinking amber light status. You can also perform a Factory Default Reset if you wish.
On your iPhone.......
Go to the Home Screen
Tap on Settings
Tap on WiFi
Look for a listing of Set Up a New AirPort Base Station
Tap on AirPort Express just below that listing

When you tap on AirPort Express, that will start up the AirPort Utility setup "wizard". The utility will take few seconds to analyze the network and then display a screen to the effect that the Express will be set up to extend the network. The illustration below is from a Mac version of AirPort Utility, but the iPhone will be similar except your devices will appear along with your network name.

Type in a device name that you want to call the Express, or you can use the suggested name
Tap Next
Watch the next screen very carefully to see that the Express is being set up to extend using Ethernet. This will confirm that the setup utility is picking up the Ethernet connection between the two AirPorts, so the Express will be set up with the correct settings for an Ethernet connection.

When you see the message of Setup Complete, tap Done and you are Done.
Now, you need to make sure that the iPhone or whichever device that you are using to test speeds is connected to the AirPort Express WiFi......not the AirPort Extreme WiFi.
If you are using AirPort Utility for iOS, you can check as follows:
Go to the Home Screen
Tap to open AirPort Utility
Tap on the picture of the AirPort Express (you may or may not be asked for the admin password)
Tap on Wireless Clients to see that your iPhone is a client of the Express. If you don't see it here, check the AirPort Extreme.....the iPhone may be connected to the AirPort Extreme even though it is closer to the Express.
Now you can run a few speed checks to see how the Express is doing. Something in the 60-70 Mbps range would be pretty good.
Post back on your results and we will continue.