You will have to replace broken micro switch in the joystick. The actual Matsushita swtiches that Hori uses in that stick are somewhat hard to find, so you might as well replace all four of them. This will result in a slightly different feel to the joystick, but won’t really impact overall playability.
Lizard Lick sells the Omron switches used in Sanwa JLF joysticks for $2.95 each. They’re nice and clicky, and will give your Hori stick a fairly tight feel. Though be aware that these switches are designed to be soldered to the JLF pcb, so you’ll need to clip off the little legs before they will fit in the joystick.
If you want something more smooth, you might consider Cherry micro switches instead, which are used in American style pushbuttons and joysticks. Ebay has a bunch available. Though again these switches weren’t designed with Hori’s joystick in mind, so you will need to trim down the L-shaped legs on the switches, or cut off the plastic clips that hold restrictor gate to the joystick (don’t worry though, the mounting screws will still hold it all together).