The following assumes that the keyboard/mouse is using Bluetooth signalling.
I don't know if this will work but I'm wondering if you had a laptop/desktop with BT whether you could 'see' the keyboard to prove that it is transmitting and receiving.
Make sure that the adapter is not plugged in and that the mouse is switched off.
This is for Win 7 but should be similar for other Win OS. Go to Control Panel > Hardware and Sound and select 'Add a Bluetooth device', When Windows is searching for your Bluetooth devices, put your keyboard into pairing mode (press the button).
If it shows up at least it proves that the keyboard is transmitting and receiving OK.
You could then try the following and see if you can make it work without using the receiver. This will be OK with a pc/laptop but at this stage will not help with a TV as it probably uses USB protocols (converted from BT signals by the adapter/receiver) to interact with the keyboard/mouse.If it works this way without the receiver then the problem is most likely in the adapter/receiver. Again this doesn't help much because the keyboard,mouse and adapter will all be matched so that you cannot replace just one if it is faulty. this is done to prevent interference with other BT devices close by.
When it shows up in the list, don’t double click it! Instead, right-click on the keyboard device and select “Properties”: In the window that opens, “Keyboard Properties”, wait a couple seconds for the services panel to load. Once it has loaded, check the box for “Drivers for keyboard, mice, etc. (HID)”. Click on “OK” to close the window. Windows will load the drivers for your keyboard. After everything settles down, your keyboard should be working with Windows.
If it doesn't show up, as I said not sure if this would work but try it if you can with another known working BT device (perhaps the mouse) and see if it can be seen this way. if you can then at least you know that the keyboard is faulty (you knew this already but at least you have proved it)
If the keyboard / mouse was supplied with the TV but was not part of the manufacturer's packaging then perhaps you may be able to purchase another cheap wireless keyboard/mouse/adapter combination as a replacement.