Need help writing a letter for work experience!

Im currently in my 2nd year at uni and I need 1-2 weeks worth of work experience this year to pass. So now i need to write up a letter and send it to a few companies the thing is ive never worked for design company before and I have never done freelance work so what do I put in the letter?

Im guessing I start off with who I am and what I want but it ends there, i plan on printing off my work to show them and im also thinking of a lil gift like a mug with 'Love Tea Love Me' as ive heard thats all I will be making on work experience. Or is there anything else I could send them that will make me stand out? Maybe create a poster for their office or something?

What you guys think about the writing and also the mug idea?
1. Get a name- don't do "sir or madam." Call up & ask who looks after the placements.
2.Do you research about the company- why do you want to intern there?
3.Tell them what you hope to achieve from the internship- is it to develop a skill, help develop a creative concept, build experience in a particular area etc.
4.Tell them why you're good- do you blog, do you design a magazine, do you organise a club night- tell them why you have the right attitude and skills to be part of their team, even for a short time.
5.Spell and grammar check a zillion times.
6.Lay out your document in a legible but beautiful way- make sure it's visually interesting.
6.Ensure you have, and can give them online access to your portfolio- a Facebook group/fan page will suffice as long as you can display your work prominently.
7. Post it, wait a week and call the relevant person, ask have they received the letter.
8. If they say no to your request, ask why- say you'd like the feedback to be successful later on.

Hope that helps.