Plan:
Open TV.
Find wires (probably just 2, I doubt it a 14", £80 TV has stereo)
Add chassis mounted switch to the live (?) wire.
Add chassis mounted headphone socket to the casing
Wire the socket to the other 'side' of the switch

This will (I hope) leave me with a TV which I can switch between
headphone and speaker modes.

Be VERY careful! Lethal voltages can be present inside a TV hours after it
is turned off. Not a job for the novice.