Is this proper English?- " I would like to speak to the Executive Director about my daughter and mine's experience at the center." Thank you!

Not quite!

"My daughter and mine's" doesn't work, and it is actually hard to rephrase it so it sounds good.

I would say "I would like to speak to the Executive Director about the experience my daughter and I had at the center." (Or, depending on context, "the experiences my daughter and I have had at the center.")

Not sure but now I really am curious about what happened at the center?

Definitely not
….. the experience of my daughter and I at the center

I think"I would like to speak to...about my experience in your centre. "

Just say "my and my daughter's"

No, “my” experience.

Close.Change mine's to my and you have it correct.

You don't need to pluralise you!.. (mines's), is incorrect in any context

So, "I would like to speak to the Executive Director about my experience and my daughter's at the centre.. Or my and my daughters experience..