Broadway Danny Rose (1984)

Danny Rose is a manager of artists, and although he's not very successful, he nevertheless goes out of his way to help his acts. So when Lou Canova, a singer who has a chance of making a come-back, asks Danny to help him with a problem, Danny does. This problem is Lou's mistress Tina. Lou wants Tina to be at his concerts, otherwise he can't perform, but he's married, so Danny has to take her along as if she was his girlfriend. Danny gets more than he has bargained for when two mobsters come looking for the guy who has hurt their brother by stealing the heart of Tina, the girl he loves.

At 04:11, there is a reference to Danny Rose managing a one-armed juggler. He appears in a party scene doing two balls in one hand.
The entire movie may be seen here.