There is an option of Wireless Bridge and an option of Wireless Repeater. Any idea if it's one of those that I need?
Neither of those is correct.
The Linksys would need to support a WAP mode. In other words it is not a router but a straight wireless access point.
But there is another solution which works on any router which is what you seem to half attempt. This is called WAN BYPASS.
Please read the tutorial here.
http://forums.whirlpool.net.au/forum-replies.cfm?t=933517
This will work for any wireless router that does not have a specific WAP mode.. which sadly is the vast majority.
So lets go back to your setup and where you left off.
I ran an Ethernet cable from a LAN port on the AE to a LAN port on the Linksys and disabled the DHCP in the Linksys settings.
What you did was half the job..
So I recommend you do it all over again though because you need to put the Linksys back into router mode. (Assuming no possible WAP mode.. which I will check on shortly).
Here are the steps from the tutorial as simply as I can lay them out.
1. You must know the IP range of the main router.. its wireless name and password..
To make this easy lets work with airport.. at default it has 10.0.1.1 as its home address and DHCP set to 10.0.1.2-200
Lets give it a wireless name of AEgen6 and password of gandalf1234
2. On the Linksys set it back to factory. Always start this from factory so DHCP is working and router is working..and you can get the computer to connect.. with nothing but one computer plugged in.. go to the setup page of the router..
Now setup these three things in order.
2.1 Change the wireless name and password to match the airport (if you want roaming network.. but this is not necessary for every setup)
2.2 Change the home IP of the Linksys to 10.0.1.254 (I recommend this address.. but it should be anything you like as long as it is outside the DHCP scope of the Airport. ).. NOTE this address is important later and will not be easy to find.. so write a label and stick it to the base of the Linksys.
2.3 Change LAN DHCP to OFF.
N.B. when you now save the settings the computer can no longer connect to the Linksys.. this confuses people all the time.. but it is fully expected.. There is now no DHCP. So once it is saved turn off the Linksys.
3. Plug the LAN of the main router to any LAN port (Not eWAN if it has one)
Power the Linksys up and with the computer still plugged in by ethernet.. make sure you get an IP correctly.. i.e. 10.0.1.xx
You can now get to the Linksys anytime you need because it is 10.0.1.254 and just type that into your browser.. so you can setup wireless after the configuration if you wanted to change the wireless on the airport for example.
The Linksys may complain about no WAN connection.. ignore it.. you are using its wireless and LAN ports.. not its router functionality.
I REALLY wish I knew the answer to this guys question! I have *almost the exact same one.
The method above works for 99% of routers.. I have had issues with a couple of obscure brands.. in which case the IP could not be changed.. hard wired into the firmware I think. Some modern routers are preconfigured for guest wireless and happen to use identical IP to the main router.. that can cause issues.. TP-Link is a brand I have run into issues with for this rather weird problem.. but you can always get around it by simply using a different IP even if it means you cannot reach the router without manually configuring IP on the computer.