That's a good question.
The type of drugs given and whether or not you refer to it as an asylum, depend on the the time and place.
As Leah said above, there are all sorts of antipsychotics that could be given and some of them do take a while to start working.
Paraldehyde, for example, can take as long as thirty seconds.
An important detail if you go with paraldehyde is that it must be given with a glass syringe because it has the unfortunate side effect of melting plastic.
It's viable if your story is set before the late eighties. These days, it's not so likely to be used.
Other possibilities from the sixties through to the eighties and nineties are chlorpromazine, thioridazine, stelazine droperidol, haloperidol, fluphenazine, pericyazine or thiothixene to name a few.
Side effects can range from extreme sedation, amotivation, cogwheel rigidity, dystonias, diarrhoea, constipation, blurred vision, photosensitivity in the case of thioridazine and...the list goes on.
Cogentin, 2 mg PRN or as a regular dose was often prescribed to counteract the side effects of the phenothiazine (all those ending in zine) and butyrophenone (those ending in idol) classes.
Some of them and some of the modern drugs can take a few days to start working properly. Drugs that take weeks are more likely to be antidepressants.
If it's an early setting you're going for and you opt for paraldehyde, an ornamental detail you might throw in is the smell.
It's really hard to describe but impossible to forget. Imagine the locked end (or high dependency section) of an acute ward. The drugs are all kept in a trolley in a room just off the nurses' station. The door to the nurses' station is shut and so is the door to the drug room. Someone cracks a vial of paraldehyde in the drug room. Within about five seconds, you can smell it through two shut doors at the far end of the day room. It's an acrid smell with connotations of flammability and some nurses in the drug room or even the nurses' station might actually get a headache.
After the patient has had a paraldehyde injection, you can smell it on their breath.