You do need a new (or at least better) wheel, and you need to push out the stump of the broken wheel stud and fit a new one. They can usually be removed easily with a hammer and pin-punch. However it is best to remove the hub before you do that, so you won't be hammering against the differential in the center of the axle.

Here is an illustration of a hub and diff. lock that looks fairly similar to yours:
[Linked Image]