What is the best way to apologize for making your partner wait?

Own it completely — no excuses, no minimizing. Lead with acknowledging how she felt, not with your reason. "You were standing there for over an hour not knowing where I was" lands differently than "I got stuck at work." The reason comes after the apology, not before.

How do I stop my partner from saying "it's fine" when it clearly isn't?

Don't accept the first "it's fine." Ask what it was actually like for her — waiting alone, checking her phone, not knowing. The real conversation starts when you show you want to hear the honest answer, not just be let off the hook.

How many times do I need to apologize before they forgive me?

Once, deeply — not repeatedly. Saying sorry five times feels like you want to be let off the hook. Say it once, mean it, show you understand the impact, and make a specific commitment. More repetition doesn't make it more real.