Congratulations Lisa! You have just made an heirloom piece of jewelry.
It has been my experience that when a bangle bracelet is created from square half hard wire, there is no need to find additional ways to harden it. As you wrapped the wire bundle, the wrap wires hardened, which will keep the main construction firmly together. After finishing the bundle, while you shaped the bracelet around either a found item or a mandrel, the base wires were hardening. Bouncing the clasp ends toward one another helps to further harden the formed shape. Most wire artists make their bracelets in the same oval shape as a human arm. In this manner, when the finished piece is worn, it shouldn't spin around easily, which could cause the bracelet to lose its shape.
Of course there are some folks who like to play it safe by putting their finished jewelry into a tumbler with a bit of burnishing compound or Dawn dish detergent and mixed steel shot to get a final shine and further harden their finished creation. The choice is yours!
Answer contributed by Dale "Cougar" Armstrong