While Bluetooth Low Energy (BLE) is indeed supported on many newer keyboards, I am not sure how this would connect to your Smartphone.
The Monitor would require more bandwidth than BLE is capable of providing.
Regular Bluetooth can do (profiles are there, mice and keyboards are there, already working on smartphones for years).
Do you know any BLE keyboard? (I am not and would guess no need and not the intention for BLE).
You could create if you like but: BLE is for "connection-less" devices and applications, e.g. every few seconds or hours a device will wake up and send something to a host (e.g. a new sensor value). A keyboard, mouse, display (which will not work anyway, except UART over BT or BLE) would need permanent connections (a real peer-to-peer connection - BLE knows the concept of "bonding", even iBeacon as complete "connection-less", not a permanent connectionn on BLE). And battery would go down (BLE is for power optimized solutions running with a coin cell battery for almost one year. Regular BT can do already what you have in mind, except HD audio and display).