How long until a sprained hand is healed? I can move it, but it's still limited. Only using synthetic elastic joint tape. Is the tape a wise thing to use?

Variable. generally 4-6wks. taping digits together is a good way to splint a joint. must work on motion. sometimes swelling takes alot longer to improve.
Up to 9 months. Hands don't get sprained, joints do. From personal experience spraining joints it can take up to 9 months for every last bit of pain to disappear,