Why didn't the machine beep when Joker killed this person?

Why didn't the machine have that long beep that happens in other movies, when the person's heart stops beating?

Most medical monitors have a "silence" and "pause alarms" feature. In this scene the "silence" setting would have been disabled but "pause alarms" setting enabled, which pretty much comes down to production design and producers wanting to have complete silence while the Joker basks in the sunlight.

Really the only reason why "pause alarms" would be enabled is if the patient was expected to die within the next few minutes (which wasn't the case here).

Also, why didn't no one come to check on the patient?

Because the monitor alarm didn't sound and it wasn't networked to a nurse's station (like the OP might be assuming).