I fucked up bad with my mother in law. How do I say sorry?

Long story short, me and my girl at the time were going through a rough time after she cheated. I was never too adamant about leaving cheaters, ‘cause I was one myself in the past and I know sometimes it has nothing to do with feelings but just a moment of lust. I decided to try and give us a chance, but I was hurting so bad and my self-steem was so low that my only way to cope with the pain was cigarettes and porn. Eventually that uncontrolled sxal appetite led to a feeling of lust towards my mother in law who is fairly young and good looking. I ended up making aggressive insinuations to her and everything got out of hand. I felt bad, told my girlfriend and ever since we’d been trying to keep going but we recently realized it’s not going to work and decided to leave it for good. I still feel a lot of guilt about the situation and even tho I know some fucked up stuff about my ex in law, I feel like I owe her a huge apology and I want to make up for my mistake. What happened is something that is just not me at all. How would you guys go about a situation like this? How would you say the biggest sorry you could?