Does a recruitment agency deserve a fee, and how much is it, and does the failure to settle this issue affect the validity of the Hajj?

If an agreement is made with an office to find a job in exchange for one month's salary from the work, this is a "Ju'alah" (a contract of reward). The office is not entitled to the agreed-upon fee unless the work is completed.

If the employer cancels the Ju'alah after the office has started working, the office is entitled only to a fair wage (ajr al-mithl) for the work it performed before the cancellation, and not the full agreed-upon salary.

A condition for Ju'alah is that the reward (Ja'l) must be known. This transaction has no relation to the validity of Hajj, and the obligation to fulfill people's rights remains whether one intends to perform Hajj or not.