When was the last time you tried using your Kindle to watch a TQS episode? I have a Kindle Fire (first generation) and was able to watch TQS shows until TQS turned over to a new system and then I could no longer watch TQS shows. This lasted a long time, I mean months; then about two weeks ago, I tried again and I was able to watch shows, so were a couple of other people who could not watch before. I wrote the technical folks at TQS to let them know that it was working and their response was "Our video company is constantly working with updates to fix these kinds of problems." I don't have all the facts of your case, but I am puzzled that they should tell you the problem is your Kindle. Kindles are automatically updated, there is nothing you have to do with them. It is more likely that the problem resides with TQS.
A few months ago, there was a thread on the Forum in which those who had Kindles were encouraged to contact Amazon to make the changes required to be able to watch these shows, but from the email I received from TQS, it appears they have worked with "someone" so that some of us are now able to watch the shows. You might want to try logging on to TQS from your Kindle and try one more time. I hope that you get a nice surprise!
