Thanks for your post and a big warm welcome with this being your first post in our community!
Sorry to hear about the issue with your TV360 remote, are you able to fully reboot the box and repair the TV360 remote to see if the issue continues please?
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Although it's a new remote you should try new batteries. It's helped other people pair the remote.
Others have had to perform a factory reset of the remote by pressing and holding the TV and Rewind button for 10 seconds.
Then pair the 360 remote holding down the 'TV' button and '0' button together until the Standby button starts to flash.
Hi again @Ghja
Make sure you have One Power turned on in the 360 settings. This should enable you turn the TV on and off.
Settings > System > One Power
On your Sharp TV you need to turn on HDMI-CEC
It's probably under something like Settings > All Settings > General. SIMPLINK (HDMI-CEC) > and set it to On.
Or it might be under Menu > Setup > Connections > HDMI CEC and set it to On.
The red button at the top right of the 360 should then enable you turn the 360 and the TV on and off.
The white button on the top left hand side of the 360 should then able you to turn the TV on and off.