I recently changed my home broadband router and now my Airport Express no longer works.
This is normal and expected, since the Express was programmed to use the settings provided by your old router. The old router is no longer there, and the Express cannot find it, so it is "lost" and does not know how to find its way.
You need to reset the AirPort Express back to factory default settings, so it will have the same settings that it did when it was new and you took it out of the box.
To do that.....
Power off the AirPort Express for a minute
Hold in the reset button first.....and keep holding it in for an additional 10 seconds while you simultaneously plug the power back into the Express
Release the reset button after the hold period and allow a full minute for the Express to restart to a slow, blinking amber light status
Now open AirPort Utility, click on the Other WiFi Devices button and click on AirPort Express
That will get the setup "wizard" started so that you can configure the Express with your new router.
Connect the Ethernet cable from the router to the AirPort Express and follow the prompts.