Might have duplicated your problem. On my Mint 18 Mate I disconnected the Apple keyboard, and the bluetooth settings software seemed to freeze. I could not disconnect the remaining mouse nor the speaker, and could not reconnect the keyboard. This persisted after killing and restarting the bluetooth settings, and even after a complete reboot of the computer. Interestingly, the bluetooth settings listed my computer's name incorrectly as "Bastien's Computer".
I opened a terminal window and ran as regular user:
$ rfkill list
Code:
0: tpacpi_bluetooth_sw: Bluetooth
Soft blocked: no
Hard blocked: no
1: phy0: Wireless LAN
Soft blocked: no
Hard blocked: no
4: hci0: Bluetooth
Soft blocked: yes
Hard blocked: no
Then:
$ rfkill unblock 4
Code:
0: tpacpi_bluetooth_sw: Bluetooth
Soft blocked: no
Hard blocked: no
1: phy0: Wireless LAN
Soft blocked: no
Hard blocked: no
4: hci0: Bluetooth
Soft blocked: no
Hard blocked: no
And the bluetooth settings went back to normal and I was able to reconnect the keyboard. The secret was getting rid of that softblocked hci0.
Hope this works for you!