What is the legal ruling on this matter, and does the first broker deserve the full commission even though the sale did not take place through him?

Brokerage is a form of ju’alah (reward for a specific task), and the broker is not entitled to the wage until the work is completed, based on the Almighty’s saying: ﴿And for he who brings it is the burden of a camel﴾. The understanding derived from this is that if he does not bring it, he gets nothing. What the seller gave to the first broker in return for his advertising and effort is a voluntary donation from him, and the entire commission belongs to the second broker.