Will a large black spider eat the spider mites in my garden?

Spiders generally eat things that are about their own size or somewhat smaller. If the potential prey is too large then it might fight back and injure or kill the spider. If the potential prey is much smaller than the spider then the spider may be unable to catch it. Spiders do not have hands. Imagine trying to catch baby chicks in a field while wearing boxing gloves. You couldn't catch anything that small, but you might get your arms around a fairly small dog.

Lady Bugs are the right size to eat such small prey items. Also, there are some predatory mites and thrips that eat mites.