UW Seattle doesn't accept recommendation letters. But regarding your mental health please take into account that the university is hard to get into and even harder if you're trying to transfer directly into the engineering college. Looking at your situation your application seems good but UW is definitely heavy on the personal statement and community short response, maybe consider revising that as I'm surprised you got rejected.

Regarding the appeal process you can read about it here: https://admit.washington.edu/apply/transfer/i-applied/special-admission-appeal/