What should I say to convince a girl to date me?

Compose your words in mind before you call her and do not be extremely straightforward. Ask few formal and casual questions and then ask her for the date. You should be cool and ask her very politely whether she would be available for that particular date or not.

How do you start a conversation with a girl in a relationship?

Approach her - Being straightforward but not too pushy is the best strategy. Just smile, say hello, introduce yourself, and ask for her name. Ask her about her day - You can strike up a conversation by asking what her day was like and prompting her to speak about it.