External Hard Drive no longer working
...any other tricks I can try myself to see if I can repair this or at
  least retrieve the data without paying for a service or software?


Based on everything you said, there are only two conditions that exist:


The USB enclosure has failed.  This is the least bad scenario.  The "enclosure" has a USB to SATA controller that is the interface of the drive.  It's not out of the realm of possibility that it "reports" back the drive, but refuses to mount/read/write/etc.  
The SSD drive itself has failed This is the worst case scenario.  If the drive has failed, there's nothing you're going to do to force it to start working.  


How can you find out which scenario is the one you're dealing with?  I wrote an answer to a similar question that addresses this situation exactly. It involves a cheap USB to SATA adapter to remove the enclosure from the equation.  If the drive works, it was the enclosure;  you get the added benefit of the adapter being a temp solution until you get the enclosure fixed (or another external drive).  If it doesn't work, then you know the SSD has died.