Trackpad physically clicks but isn't responding

The Click action is achieved via a physical blister switch which sits on the back side of the trackpad

So if the trackpad is not physically moving you likely had junk jamming the edge preventing it from moving. If the trackpad is moving then either the adjuster needs adjusting or the trackpad its self has failed. I’ll warn you its a big job! And I often also take the time to replace the battery as well as I’m sure your’s is getting to the end of its life. Let’s get a better view of things with your battery, install this gem of an app! CoconutBattery take a snapshot of the apps main window and post it here for us to see Adding images to an existing question

In any case here’s the needed parts:

• MacBook Pro 13" Retina (Late 2013-Mid 2014) Trackpad
• MacBook Pro 13" Retina (Late 2013-Mid 2014) Battery

Here’s the guide to put both the trackpad and if needed a fresh battery MacBook Pro 13" Retina Display Late 2013 Trackpad Replacement

I would follow the guide removing the trackpad so you can clean the edge of the case using some kitchen cleaner and a paper towel (wetting the towel) getting the grit and gummy mess off. Do the same on the trackpad edge as well but you want a fairly dry towel as you don’t want the cleaner to wick into the layers of the trackpad.

To adjust the click plate, carefully rotate this screw a quarter of a turn at a time as you don’t want to allow the screw to come out. Test things by pressing the Trackpad until you here or feel the click action of the switch.