What you have here is less a legal issue and more a matter of persuading a lawyer.
"Pro bono" cases are generally taken on at an attorney's own discretion; in order to acquire pro bono representation, you quite simply need to convince the attorney to take the case. The first step in this, and likely the only thing a letter will accomplish, would probably be to get the lawyer to interview you.
Attorneys' time, especially if the attorney is experienced, is valuable, and you're effectively going to be asking the attorney to donate what will likely be several thousand dollars worth of time to your cause. This doesn't mean that getting pro bono representation is a lost cause (far from it), but you might try to explain the following points, in no particular order:
1) The usual solution for indigent defendants is a public defender. They're often extremely skilled and experienced, if badly overworked, and Seattle has a crop of particularly good ones. Why do you need a private practitioner to represent you pro bono instead of a PD?
2) What is the basic nature of your case? (Keep it short. The attorney will get the details at the interview, if there is one.)
3) What makes your case "special"-- that is, why should the attorney take your case? (This is going to be virtually any reason that a reader would find persuasive.) "Pro bono," as you may know, means "for the good," in Latin. How would the lawyer be acting "for the good" by taking your case?
Keep it honest, keep it persuasive. You are quite simply trying to talk a professional into helping you for free, and your task is both as straightforward and as difficult as that.