Father (72M) confessed his affair to me (45M) on his death bed. Do I tell my mother?

My (45M) father (72M) recently passed away from health problems, and he had one last request of me, which he asked me to handle with discretion. When he died, there was someone he wanted me to call and inform… and he hoped I could keep it between us.

I got as much of the story as he was willing to tell me:

He was an officer in the military, and he spent a year deployed to Korea. When my sister and I were kids (back in the states with mom and our grandparents), he met a woman there. She was the daughter of a wealthy family; they met through a social circle, and eventually fell in love.

My father told me that while he cared deeply for my mother, Jiyeon was the actual love of his life. He regretted everyday he wasn't with her. Apparently, as his time there was ending, he had plans to divorce my mom and be with her. But he ultimately decided to break off the affair, for the sake of my sister and I. He never told a soul until now.

I went to the toolbox in the garage he told me about, and sure enough there were some old polaroids (she was indeed quite the looker) with writing/dates, and some contract information.

I reached out and told her my father had died. She is retired from a long career, and had a family of her own now. She said that while she truly loved my father, they were young and stupid, and she deeply regretted bringing this burden to me and my family. She apologized, sincerely thanked me for telling her, and wished me well. This may sound strange, but this conversation was a nice form of closure for me, someone who never felt like I knew my dad on a personal level very well.

It wasn’t easy for me to do, but it felt important for me to carry out his last wish. He had done everything for me in life.

The next part (not telling anyone else, including my mom), I feel very conflicted about. I’m very sad from the loss of my father and want to honor this wish, but I truly feel like my mother and sister have a right to know. But also part of me doesn't want to tarnish his image, and bring my mother unnecessary sadness at this point in her life. What should I do? Should I tell them? How would you go about this?

tl;dr my father confessed his affair to me on his death bed, but didn’t want me to tell anyone else. But I feel the need to tell my family about it.