Bopeep, IDK if you're still reading here or not. But if you are, I want to say that I totally understand how you feel about your mum. I would too if I were in your situation. And, IMO, it's noble of you to put her needs ahead of your own.
IDK if your letter will work or not. I hope it does, but I know it might not. As some posters have suggested, ES may resent what he sees as an attempt at "shaming." But I get that you felt you had to try, and at least, you'll know you did.
Please realize there is a third possibility. While ES may not reach out to your mum right away, he might, at least, begin to think about it. I don't think she should try to make contact any more though. I don't think more cards/letters than she has already sent will get her any results if the others haven't. But that's her call, of course.
Be that as it may, IDK if there's much else you can do to make your poor mum feel better about this situation. Have you tried finding ways to help her keep her mind off of it? To help her shift her focus to those parts of her life that bring her satisfaction and joy? That might be the best you can do for her in case ES doesn't reconnect with her or while she's still waiting.
IMO, you are a good, loving DD. I hope your efforts on behalf of your mum bear fruit.