I'd never ask for a girl's number, most of the time I just say "Shoot me your number, let's text". But the key isn't about how or what you say, it's all about timing.
Here is the golden rule Number close when her level of interest is at peaked
This can be 2 responses in or 3 min into the conversation. With time and experiences, you'll get better at getting the timing down. With that said, you have to create a strong enough emotional connection. Your conversation was mostly fluff talk. I'd have number closed after she said "Let's"
Her: Let's
Me: Wait like friends that actually text back and hang out with each other?
Shoot me your number. Let's text.