We had an appointment but my friend didn't meet me on time. I want to know why she didn't.

Use the noun before the pronoun: My friend and I had an appointment, but she didn't show up on time. I want to know why she was delayed. Being on time is one of the most important attributes.

we had an appointment but I didn't meet my friend on time. This sounds like you were the one who was late. I want to know about why she has delayed. From this sentence it seems she was the one who was late? being on time is one of the most important property of a person. Note that saying she was 'delayed' suggests that she was late for reasons outside her control - perhaps there was a traffic accident, or she couldn't get a taxi. Assuming it was she who was late and not you, another way to write this is: I was scheduled to meet a friend, but she was late. I want to know what delayed her, as punctuality is very important to me.