The first question I would ask is "what is the problem" or particularly what is the root cause issue? You've identified a lot of tools, but have indicated that they are not enough, and you're probably right. But then I'm wondering, "if these solutions are not "right", then what is actually wrong?" Is this a technology issue, or is it a management issue, or a training issue or a service model issue?
Without knowing anything more than what you've shared, I would suspect that the issue may be that staff non-responsiveness to queued requests is indicative of staff not having clearly understood protocols of who is supposed to respond to what requests (e.g.- service model and/or management issue)
One solution may be to assign a service "quarterback" to manage the dashboard and assign service requests to designated floor staff (Perhaps set-up "zones" in which on-duty staff members are responsible for responding to service requests that arise). For communications, you could assign service staff with headsets (e.g.- secret service) to monitor and receive assignments.
I guess the real solution would arise from having a better understanding of the root cause of the problem.