Thanks for the replies.

We had tried to call the two companies recommended by locals many times. They were either busy or had put us on hold for 20', after which we gave up. I think the two major companies are the same place with different numbers.

We mentioned this to some people we met after lunch, who had a similar experience because it is Fashion Week--no taxis available.

After that, both my wife and I tried to call and got through, but the dispatcher hung up on both of us. We saw some empty taxis around town and my wife had the great idea to approach a taxi driver to call the dispatcher, which worked. At least we are booked; we'll see if a taxi shows up in the morning.