Welcome to the American Justice system. In the Feds, you only have 14 days from sentencing to appeal a sentence. It used to be unlimited, but that was clogging up the courts because too many convicted people were exercising their constitutional rights, so they took it down to 14 days (or 30 days if you are the Government prosecutors because some animals are more equal than others).

https://www.justice.gov/jm/jm-2-4000-time-appeal-or-petition...

A lot of those 14 days get taken up with time in holding cells being moved from the Courthouse to a Federal Detention Center (or contracted local jail), moved from the pretrial holding facility to your final place of incarceration via Diesel Therapy, etc. If you are lucky you will get access to a law library, paper pad maybe an hour a day, but the Supreme Court has rules that is enough to fulfill your constitutional right to 'access to the courts'. You also need to figure out how to get the law library to weight your mail to put correct postage and get the certified mail forms, and to the jail/prison mail room during their 'open house' hours (normally 1-2 hours and at the same time lunch is served so you might not be eating lunch that day if you are at a place with controlled movements) so that you can send legal mail to mail your appeal (and how to come up with stamps to send certified mail to the Court and Prosecutor).

Edit: Getting stamps includes waiting for the once weekly commissary day to order them, and then potentially waiting for commissary to be delivered to your cell (shucks, it's delivered after the mail room open house 1-2 hour window) so that's up to a 9 day loss of time just to get stamps/envelopes after the appeal is written in triplicate (Court, Prosecutor, personal copy) so you can weigh it and know how many stamps you need to purchase.