It's quite possible that you just broke the shaft at the hub, If so, it's easy to replace. Worst case, you'll have to take the hub off. But when mine broke, I was able to grab the end with my fingernails. It slides out.
More important is what you replace it with. Bugeye axles are notoriously weak. You can get axles from a later Spridget, which should be OK, or racing axles if you are really concerned about it. The replacements I used were the BTA806 type; these are a pretty common replacement. There are others, though; do a search and you'll find a lot of postings about this.