iPhone 5S, LTE does not work
I assume you should try to test the SIM with another phone in order to see if it really works.

Test

All you have to do is pretty simple:


Get your SIM ejector and take it off from your current iPhone
Get another iPhone and put it in
Unlock the SIM on the new iPhone and make sure LTE is activated properly under Settings > Cellular > Cellular Data Options


Once done this, I'd definitely try to contact your carrier's support (which if I'm not wrong is AT&amp;T in your case) and report them the result of the test and see how they can help you.

Hope this helped you out, and don't forget to let me up to date on the question!