I have rehabbed several Coach products that are made of (or lined with) the 65th Anniversary Legacy Striped sateen fabric, like your wristlet. They look delicate but they are tough! They hold up very well and rehab nicely.
You could put that wristlet in a lingerie bag (or tie it up in a pillowcase) and cycle it through the washer with Woolite. But if you prefer you can handwash it in the sink with Dawn and scrub it with a soft brush, like a nail brush.
Arrange it nicely on a towel to dry so that it doesn't get wrinkled and apply some leather conditioner to the leather parts. When I am done I always spray cloth bags with a fabric protector to help keep them clean, I use Apple Rain Garde.
See links below for before and after pics and info for one of my toughest Legacy Stripe rehabs, the large Katy Travel Tote from 2006, it was in bad shaped and needed a lot of work but it came out well. Your wristlet looks to be in good condition, mostly dirty, so it should clean up nicely. You could use acrylic paint to restore the gold finish on the leather if you wish.
Is your wristlet missing the Coach hangtag? You can call Coach Customer Service and ask them to send a gold metal hangtag as a free replacement.
Katy Tote Before:
https://forum.purseblog.com/threads/coach-rehab-and-rescue-club.833400/page-982#post-31650475
Katy Tote After:
https://forum.purseblog.com/threads/coach-rehab-and-rescue-club.833400/page-982#post-31650503
It's my namesake bag so I had to try and save it! You can see a picture of a Legacy Stripe Wristlet here, this one cost $58 in 2006:
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