I’m trying to print wirelessly from my iPhone5s to our Brother printer HL-2340WD

That printer does not support AirPrint→About AirPrint - Apple Support; it isn’t on the list in that link. If you have a Mac you can get an AirPrint driver app such as Printopia, or check with Brother, they may have an iPhone driver app for it.

As long as the printer supports AirPrint (About AirPrint - Apple Support) and is connected to the same WiFi network as your iPhone is on, it should see that printer and print to it. There are no drivers or anything to install with AirPrint.

Ah, I see the HL-L2340WD is AirPrint but I guess that’s a newer model?

The 2340DW is listed, but not the 2340WD. Unless the original post contained a typo 😁