General :: Bluetooth Doesn't Work?

We have several android devices in the family and only one bluetooth mouse/keyboard. we prefer joining the tablets with hard peripherals when not on the go.

Here's the problem: one of the tablets' screen is burnt out so I have it plugged into a television via hdmi. I can look at the tv to estimate where to touch the tablet to sync up the bluetooth peripherals, but it's tedious and takes too long. We're swapping the bluetooth peripherals constantly so it would be better if the tablet "remembered" the bluetooth devices and didn't need to be paired each time. I realize that bluetooth doesn't work that way, but there has to be a way to do this, or a better way at least.