LED Replacement for T5 Fluorescent lights
There are three ways to go. 


Fit T-5 LED retrofit "tubes" into your existing fixture. 
Purchase a new fixture which has socketed light bulbs, and fit LED light bulbs in the sockets. 
Purchase a new socketless fixture which has LEDs soldered down.  Socketing LEDs is ridiculous since the LED emitter will last longer than the socket! (not so high hopes for the electronic driver circuit; that's probably what will die actually, but quality varies dramatically on that.) 




Most of us here will recommend #1, for sheer simplicity of installation.  In this case, you obtain LED retrofit "tubes" (they look like a T5 fluorescent tube, but are festooned with LEDs instead of filled with argon gas or whatever). You must be conscious of what you are buying, because there are two ways to go here: 


Plug-and-play, which means they pop right in where the tube went for ease of upgrade, except they still require the old fluorescent ballast (an electronic module) to be present and maintained. If the ballast fails you must maintain it.  Upside; you can always roll back to actual fluorescent tubes.
Direct-wire aka ballast-bypass, in which you pop the ballast maintenance cover off the fixture, and rearrange the wires so the AC power feeds the lamp sockets directly. Then the fixture only works with LED "tubes". 




The fourth (first, really) option is to replace the fluorescent tubes with new actual fluorescent tubes.  Do this if you see black  rings near the ends of the tubes, or other signs the tubes are worn out.  If the tubes look healthy, the ballast may have failed - and that means don't do plug-n-play LED!  T5 tubes are expensive enough that I wouldn't want to buy some just to find out.  In 1-2 bulb quantity, LED "tube" replacements are cheaper than plain tubes!