How to write a really short letter of intent?
Carefully address the question as phrased by the institution.  Different schools may describe what they want as a letter of intent in slightly different ways that may help you understand what they're looking for.

Broadly speaking, a letter of intent is not about your past.  It should explain your intentions for the future.  (Stuff about your past is probably better in a personal statement, if you're asked for one.) A letter of intent should answer the question, what will you do if you come here?

In any sort of admissions or employment situation, whenever you're asked about the future, a good answer usually includes some specificity about your objectives, something testable, what you'd like to work on, how you think you might proceed toward your goals, etc.  Hope that's helpful.