I just have a little of my own experiences for you to use as a guide.
I have been asked out via letter before and always have found it awkward. I think letter format allows a little too much expressiveness in a situation where a positive response is uncertain.
If you do write a letter, keep it brief and UN-intense.
When asking someone for a date, whether for a dance or to coffee, you want to give yourself and them an easy out. You want to sound like you like the idea of taking them, but won't be too disappointed if she can't or won't go with you. I recommend the verbal approach. Among its advantages:
1. With acquaintances, there is something attractive about a person who is only mildly interested in you. It is non-threatening, requiring no commitment and creating no implications. Verbal approach can appear to require no prep and little thought, so it can look very casual.
2. You can modify your approach as you watch her response (something you can't do in a letter).
3. There is always risk involved in asking someone out, but if you do it verbally there will be no physical evidence for anyone to tease about later.
I suggest something casual, along the lines of this:
"Nice shoes"
"Thank you"
"I just had to get some new shoes for that dance coming up in May. I don't know whether you're going or not...[slight pause for her to register new direction of conversation] but if you are I'd be happy to take you." Then a little half smile like it sounds nice but your life doesn't depend on it.
If she raises her eyebrows with no smile like she is surprised or says "I don't know..." she may not wish to go. That is a good time to gracefully shrug and say "Well just let me know. We'll either go together or we'll see each other there, so it's kind of all the same, right?" And smile. Then go head somewhere else so she can think about it by herself. This encounter is breezy and minimally awkward, plus it's pretty much over unless she decides to tell you yes. If she doesn't say anything more on the subject, that's a no. If she tells you she does want to go with you, Yay!
With a letter, she has to either approach you to discuss it (which is only positive if her answer is a yes. A "no" conversation will be uncomfortable for both parties), or she will never bring it up and will avoid you because she feels awkward about not wanting to go with you and doesn't want to have to tell you that's how she feels.
This is just one girl's advice, and I'm just me! It's the best I can offer, but this girl may not be like me, so it's hard to say. If she makes lots of eye-contact with you and smiles a lot at you, those are good indicators that she likes you, at least as a friend.
Now decide on your plan, and BE CONFIDENT! If this girl says no, it's no skin off your nose. If she says yes, WOOHOO!