Space miners are only for asteroids.
Constructors are only for Starbases. To "Mine" those resources you need to build a starbase on it. So, it requires a constructor, and not a space miner.
Constuctors also build economic, military and influence starbase. I don't think you were asking that but i threw it in to avoid confusion.
To get more out of your space miner, you need the Space Mining techs, that you said you alreday have.
To get more modules to place on the resources you need to reserach the industrial techs as Burress said. And of course you would need to send a sperate constructor to the resource for each upgrade.