I think it's safe to say that anyone who's been transcribing for any length of time has had the experience of missing a deadline at least once. Whether it's because you oversleep and lose valuable work time, end up with a sick child to take care of or realize your audio sounds like it's been run through the blender and will take twice as long as you thought, it's bound to happen at some point.
The question is what to do when it happens.
The most important thing to remember is that communication is key! Whether you're working directly with a client or contracting for a larger transcription company, it's important to conduct yourself as a professional and keep them updated at all times.
Once you realize that you're going to be late, drop them a note and let them know that you'll be late and give them an estimate for completion.
It is not necessary to launch into a detailed account of why you'll be late, but if you have a good relationship with your contact, I think it's okay to include a sentence about the why. I know there's debate about this, and I'm not sure there is a right answer, but an account representative I asked said that she only needs to know the why if the issue has to do with the file or something on her end.
Finally, once you know you're going to be late and have sent an email letting your contact know when to expect the transcript, make sure you hit that deadline!