To fix a broken handle on a regular mop, buy a replacement handle that fits your mop at a hardware store, and follow the manufacturer's instructions on how to install it. Some handles clip onto the mop head, while others simply slide onto them.
The most common problem with Shark steam mop handles is that the plastic part of the handle breaks off the metal part. To fix this, use a utility knife or utility shears to disconnect the internal stems in four places where they are connected. Make two holes in the plastic handle opposite the existing ones. Put aluminum spacers in place, and bore out the binding posts. Set spacers inside the metal handle to keep the binding posts from moving when they are put in place.
Put the plastic handle over the metal one, and insert the binding posts. When putting the binding posts in place, first insert the stems of the binding posts in the holes in the plastic handle, and then slide them further into the aluminum spacers. Put the threaded sections of the binding posts into the new holes you've drilled in the plastic. Tighten the binding posts using a screw driver.
Aluminum spacers that are available in hardware stores are usually smaller or larger in diameter than the holes in the metal handle. Smaller spacers are usually suitable for the task, but if you're concerned that they are somewhat loose, use an expanding adhesive, such as Original Gorilla Glue or Epoxy. Apply the adhesive onto the spacer, and insert it in the hole.