Assuming that Mike has a padlock to which Jane has a key: Mike uses one of his padlocks to lock the lid of the box to Jane's padlock. He then uses another one to lock the box base to Jane's padlock. Jane can now open the box by opening her padlock, thus disconnecting the two Mike-padlocks (which are still attached to the lid/base respectively).

EDIT: This approach is actually used in real-life scenarios where multiple persons must have access to something without sharing a key: They each provide a padlock, and the padlocks are chained together, ultimately connecting the two parts that must be locked together.

Assuming that Mike has a padlock to which Jane has a key: Mike uses one of his padlocks to lock the lid of the box to Jane's padlock. He then uses another one to lock the box base to Jane's padlock. Jane can now open the box by opening her padlock, thus disconnecting the two Mike-padlocks (which are still attached to the lid/base respectively).

EDIT: This approach is actually used in real-life scenarios where multiple persons must have access to something without sharing a key: They each provide a padlock, and the padlocks are chained together, ultimately connecting the two parts that must be locked together. http://www.spurgeonworld.com/blog/archives/2006/12/daisy_chains.html

Assuming that Mike has a padlock to which Jane has a key: Mike uses one of his padlocks to lock the lid of the box to Jane's padlock. He then uses another one to lock the box base to Jane's padlock. Jane can now open the box by opening her padlock, thus disconnecting the two Mike-padlocks (which are still attached to the lid/base respectively.

Assuming that Mike has a padlock to which Jane has a key: Mike uses one of his padlocks to lock the lid of the box to Jane's padlock. He then uses another one to lock the box base to Jane's padlock. Jane can now open the box by opening her padlock, thus disconnecting the two Mike-padlocks (which are still attached to the lid/base respectively).

EDIT: This approach is actually used in real-life scenarios where multiple persons must have access to something without sharing a key: They each provide a padlock, and the padlocks are chained together, ultimately connecting the two parts that must be locked together.