Bigger drawer. Think laterally.
Make a hook to hang it on
Alas, it's not my drawer, and the manager may object.
Or electronically tag them to stop people walking off with them after work like they do in prisons. Though a governor did once some how manage to go home with his set. Meaning the whole prison had to have every lock changed.
I think that electronic tagging may be considered an invasion of privacy here. Besides, it'll probably just show it going to my apartment or the managers house; we're both terrible for losing keys.
This happens more than you'd think. At least one trade union for social workers here offers "Insurance against losing master keys" specifically as a benefit of membership, showing that a lot of us lose keys, or are a bit paranoid about losing keys...