Is a partner in a law firm obligated to share with his partner the brokerage profits he earned from work unrelated to the legal profession, especially given that the partner expended no effort in this work?

The criterion for determining a partner's entitlement is the condition agreed upon in the partnership contract. A partnership of bodies (sharikat al-abdan) may be restricted to a specific type of work (such as legal practice only), or it may be absolute, encompassing any work. If the agreement was restricted to legal practice, then your partner is not entitled to anything from the brokerage work. However, if the agreement was absolute, then he has his share in the brokerage work and other endeavors.