You answered your own question, if you are an asset to a business you already know the proper course of action.
Anything other than walking in and saying “I fucked up, lost the money. (reach into wallet) so here is the money I lost, for my sake and the companies I will pay more attention.” you have no other course of action that will make you look half way decent. BUT you should have done that BEFORE it was found missing. Now it still looks questionable.
Last thing you want to do, also, is burn the bridges with them if you want to use them on a resume.