Dear Sir/Madam,
I won't waste your time with a long cover letter. I have good experience in making transcriptions, since my work as a journalist is mostly listening to several hours of recorded interviews and writing down everything I hear, which requires a lot of attention and accuracy. I also have experience from working as a Transcription Freelancer on oDesk. I believe my skills would be ideal for your project, I am used to work with deadlines (I never failed to meet one) and I would like to work for you.
I am new here, but willing to work hard to build my reputation (that's why I am quite cheap to hire, without giving up the quality of the outcome). If you want to, you can send a piece of that audio to me for a short test. I will work with that and we can continue afterwards, but only if you are happy with the result.
Thank you for considering me for this job!