Is it permissible for me to demand a certain percentage of the recovered money if this was not agreed upon in advance, and is the money taken with the help of a relative - which might cause embarrassment to the other party - considered lawful?

The recompense promised to you by the owner of the stolen money, based on the concept of ja'alah (reward/stipend), requires knowledge of the designated reward. Some jurists have stated that ignorance regarding the specific ju'l (reward) does not invalidate the ja'alah if it does not prevent delivery, such as when one says: "Whoever returns my slave shall have half of him." Agreement on a specific percentage of the recovered money prevents dispute. If this is not achieved, the ja'alah becomes invalid, and in case of dispute, the ju'l al-mithl (customary equivalent reward) becomes obligatory, as Ibn Taymiyyah said: "The compensation in invalid contracts is similar to what is customarily and ordinarily due in valid contracts." It is permissible for the commission to be a percentage of the amount to be collected, and there is no harm for you in claiming it now, either yourself or through someone else.