This answer is applicable to all Miatas from 1990-2005. The power antenna is controlled by a motor that pushes and pulles a nylon, ribbed "cord". It is common that the nubs on the rope will wear down so the motor can't grab it.
You need to determine if the actual motor is broken, or if just the mast is broken.
Look very closely at the black, plastic ring around the antenna. You will see two small holes. The ring is simply screwed in, but you'll need to purchase some cheap, snap-ring pliers (about $7 USD) to remove it. This will allow you to remove the entire trim ring and work.
If you only need to change the mastOnce the ring is off, power the radio on, so the motor is "pushing" the antenna up. You can then pull up on the mast and pull the hole antenna out, including the nylon cord.
There is a metal sheath around the base of the antenna. You'll need to remove this and replace it on your new antenna.
Stick the new antenna with nylon cord into the antenna hole. Have a friend turn off the radio, while you gently bob the antenna cord, to "fish" for the power antenna's gear. This may take a few tries. Be prepared to carefully install the mast once you find it's pulling!
Once it is in, replace the trim ring.
how do you run the wire for the power windows on a 91 Mazda miata
The 2011 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2004 Mazda MX-5-Miata is 5 ft. 6 in. (66 in.)1 one-touch power windows wide.
The 2001 Mazda MX-5-Miata is 5 ft. 6 in. (66 in.)1 one-touch power windows wide.
The 2014 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2006 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2012 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2008 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2005 Mazda MX-5-Miata is 5 ft. 6.1 in. (66.1 in.)1 one-touch power windows wide.
The 2013 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2010 Mazda MX-5-Miata is 5 ft. 7.7 in. (67.7 in.)1 one-touch power windows wide.
The 2002 Mazda MX-5-Miata is 5 ft. 6 in. (66 in.)1 one-touch power windows wide.