Cheap alternate solution: Mike closes the padlock through the ring's hole and sends it to Jane. She won't be able to actually wear the ring until they meet up in person and Mike can re-open the padlock with his key, but she will nonetheless have the ring.
Whether this is an actual solution depends on (a) how loosely you interpret the goal "get the ring safely to Jane" and (b) whether a Kleptopian thief would steal a padlocked ring, knowing they would have to ruin the ring to get it free of the padlock.