With a murdered employee and a missing patient, that hospital would've locked down like the Thunderdome while the Bride was still lying in Buck's truck and staring at her feet. Regardless of whether they thought she was responsible (and they probably wouldn't have -- they'd find it way more likely that some lunatic had killed Buck and hauled her comatose ass off like a sack of potatoes), the cops would be covering all exits and combing the building looking for her.
"OK," you think, "but maybe it wouldn't occur to them to look in the parking garage. Maybe they'd be too worried about whether there was a killer loose in the building to go see if Buck's car was missing." Well, sure, we could buy that for an hour or two. But once they'd established that the killer and coma patient didn't seem to be in the building, they'd need to figure out how to track their asses down. And someone (either the cops or a super sleuth doctor) would definitely notice that Buck's keys were gone, because Buck's keys look like this:
And the vehicle they operate looks like this:
The cops don't even search it anymore. The Purell costs were breaking their budget.