Well, try to check your downloaded file. The "update_kindle_5.3.4.bin" should be 171598529 bytes long and its md5sum should be "579c8d8f6f38111e1e02105349505b42" (or its sha1sum which should be "9c87da5c32acc94ccf433f198f18606fcab6201b").
I assume that when you say "tap the update device button" you actually mean a two step procedure. First you enter "Main Menu" -> "Settings" and then when the "Settings" screen is displayed you enter "Main Menu" -> "Update Your Kindle".
And I hope you read:
Transfer & Install Software Updates Manually
http://www.amazon.com/gp/help/custom...deId=201177270
Troubleshooting Kindle Paperwhite Software Updates
http://www.amazon.com/gp/help/custom...deId=201177280