How can I make a kit that has Unbreakable Items in it?

Solution 1:

Instead of using the unbreakable attribute, use the ench attribute to add almost infinite unbreaking to the item. (It is still possible to break it, but it would have to deal the damage of killing about 130,000 ender dragons) The command is /give (Name) minecraft:(item) 1 0 {ench:[{id:34,lvl:10000}]}.
The item now has unbreaking level 10,000.

Solution 2:

By the command it seems like you're using a plugin. If you have essentialsx on your server, which is a pretty common and good to have plugin (in addition to its capable kit system), the following kits.yml is valid and works as expected.

kits:
  tools:
    delay: 10
    items:
      - stonesword 1 Unbreakable:true