If you could give me one piece of advice as I ask someone to forgive me for what I have done, what would that be?

For one and only one piece of advice, I would say this:  Once you have asked for forgiveness, please be patient with the person who was hurt.  Do not expect instant forgiving from that person.  Asking for forgiveness requires a humble approach and letting the other person choose when it is the best time to forgive.

For additional information, see Learning to Forgive Others.

Is there anything I can do to encourage my brother to forgive me?

Did you apologize?  Did you show him that you are aware of your error and have taken steps not to repeat it?  This may help him establish trust in you which may help him to forgive you.  You will need patience as he makes up his own mind.  Your trying to put pressure on him to forgive will not be helpful.  He needs to see the value of forgiveness and willingly say yes to it.

For additional information, see Learning to Forgive Others.